They can't.

A modern day army, with that kind of strength, would wipe out any main (police) resistance pretty fast. Assuming they prepared it somehow, and not just you know, discover that there's some land there and have no idea how to go around.

The main reason for it, is not necessarily the lack of army, but mostly the lack of equipment.

Let's see how that could go down:

1. The invasion begins (and they did not use air planes, or artillery to cut any communication mean, just because.


2. The government gets to hear about it. And decides to mobilise the troops to resist the invasion (because they can be irresponsible as well).


3. They somehow contact the population (with the communication that still wasn't cut, remember?) like with SMS/Email.


4. People rally to the designated centres (because they are patriotic/suicidal) and somehow the invading army does not get to know about that, observing many people moving to a specific destination.


5. They give them the best they have: a set of police equipment: rifles, guns, some helms, etc. (This is assuming, their police was well funded and had that much surplus).


6. They train them (meaning, they give them a chance to shoot, if they don't die trying, then good enough, they can join the front).


7. The first mobilised troops get to face the invasion (who were kind enough to wait for them to come up).

They are now equiped with some shields, light guns, and maybe gas grenades... to face tanks.

The Russian (in)famously faced the German army on such similar imbalance. But first they needed time to gather their troops, and had to fall back a few thousands of km first. On Iceland's scale, that's the sea. And they made it work by having a much more populated army. Iceland has 380,000 inhabitants. You don't get the 60-80,000 stronged-troup in 2 weeks. And modern weapons are slightly more effective than WWII's.

Altenatives

I can see two alternatives.

• They were in the situation you described, realise that they have no chance, and surrender. However they try to arm and organise some resistance. When there's such a strength imbalance, guerilla's tactics are the go-to choice.




• They were not completely irresponsible and had a plan put in place to defend themselves. Which, I guess most evolved countries do. Either they train their citizens regularly (like Switzerland), or they have agreements with stronger countries, etc. And put the plan they had into place.

Long and short, two weeks isn't really long enough for many plans. We should have been gearing up to defend ourselves before anyone ever arrived, but we can't change the past.

In the position of the not-icelandic government, my first move would be to hire a substantial paramilitary mercenary group to help defend my nation. Our biggest military resource right now is probably our wealth.

My second move would be to form a militia and arm them. If there aren't enough weapons to go around, that's a limiting factor since any competent invader will have blockaded us. As other answers have noted, Not-Iceland has a heck of a lot more people than the invading army, if even a modest percentage can be armed and equipped and are willing to fight, that's an army of similar size even if it lacks in quality.

Thirdly, I would also be making serious overtures to anyone who might be sympathetic and powerful enough to help out.

Guerrilla Warfare with the support of the population has a long history of success. Take Finland during WW2 as an example. With the assistance of a bitter winter, knowledge of the terrain and a tenacious determination to drive the invaders out, they repelled both the German and Russian armies using a militia armed with hunting rifles.

Same way small countries without standing armies do now. The police are used and added to. Many small nations police are specifically sent on peace keeping missions so that they can learn some military type skills.

Would the defending country's government be able to appeal to the UN or NATO for support/an expeditionary force?

They could appeal to anyone they want or pray even. But it's totally dependent on political expedience and economics on whether they get any help or not. Iceland being basically part of Europe would get quicker more effective support than somewhere like Tonga. PNG has been appealing to everyone in sight, yet despite being next door to Australia receives no help from an invading modern army practicing genocide.

In practical terms without treaties with other countries they're pretty much on their own.

How can a country create an army in two weeks when faced by invasion?

They can't.

An army implies a great deal of organization, training, and support. There is no way you are going to pull one together in two weeks. You need tactical & strategic leadership, command & control structures, logistics corps, and people that understand and accept their place in the overall structure along with many other things to have an army. There is no way to put all of that together on the fly while an enemy is advancing across your territory.

Can a country without an army mobilize for an effective defense in two weeks?

Maybe.

This would hugely depend on the social/political structure of the country and populace, along with certain aspects of their economy and geography.

Guns

You mention limited private ownership of firearms. This is a huge damper on their likelihood of success. People who have never handled a gun tend to be afraid of them. They don't know how to carry, load, ready, aim, fire, handle recoil, clean, or otherwise maintain their weapons. They may have seen things in the movies, TV, or games, but there is a huge difference between what generally comes from the entertainment industry and reality. It is generally accepted (at least where I learned) that you need ~100 rounds through a firearm to get accustomed to how it handles. That does not mean proficiency, just that you are generally hitting in the vicinity of the target, can reload in a reasonable time frame, and are unlikely to hurt your self or panic/drop it when it goes off.

You mention 20,000 invaders. Just to match them in numbers you need to come up with 20,000 people, 20,000 firearms for them, and 2,000,000 rounds of practice ammunition, and time for them to do said practice before they are even close to being ready to handle a weapon without direct supervision. Not to mention the logistics of making sure the ammo matches the weapons for any given group. Even if you managed to pull it off, you are still at a huge disadvantage in this respect because your people just barely know how to use what they now have, your opponents will likely have fired thousands of rounds through their weapons, as well as having cross trained on the other weapons their forces use. A handgun handles very differently than a rifle or shotgun, and those handle differently than each other.

By contrast, a culture where the majority of the citizenship is familiar with and has ready access to firearms stands a much better chance. Particularly one in which marksmanship is prized and striven for. In such a place people will know how to handle their guns and what their strengths/limitations are. They will have their weapon of choice where they can get to it quickly and if they don't have ammunition readily available, they know what they need. They are likely to have practiced enough to hit targets reliably, and they will know how to care for their weapons. It is also likely that they have practiced with multiple types/styles of firearms at some level and could quickly be trained on a new weapon if their usual choice was unusable/unsuited for some reason. On average, they would be a decent match for the soldiers in terms of weapons handling and marksmanship.

Intelligence & Communication

The biggest part of war is not throwing bullets at each other, it is information, and making sure it reaches the right people at the right time.

• Where is the enemy?


• What do their forces consist of?


• What are their objectives, both short term and long term?


• Who do you have available to confront them?


• How are those people equipped?


• What is their level of experience?


• How can you pass orders to the people in the area?


• How will you get feedback and adjust when things inevitably change?


• Can you keep communication lines open when the enemy is trying to close them?


• Can you keep the enemy from listening in on your communications?

This is just a sampling of the types of information that needs to flow through a military organization for it to be effective. Your outlined scenario doesn't offer much in the way of hope on any of these points. Particularly with the last two, communications that are resistant to disruption and interception generally require some special equipment and arrangements to be in place before you can use them. It would be difficult to establish such after the invasion has started.

The biggest thing that would help is to have multiple redundant communications lines connecting your population centers. By this I mean no single points of failure in the network as a whole. Each town or city must be connected to several others, so that information can flow around any nodes the enemy takes offline. There needs to be a mix of mediums involved as well. Land lines and fiber can be cut, radios can be jammed, messengers can be intercepted. Doing all of these reliably at the same time is unlikely.

Command & Control

How are you going to pass instructions to your people and have them relay information back to you? You need the 200 militia in area X to form a blockade and delay the 1000 enemy soldiers that are coming down highway 1 for a day so you can get other people in position to stop them. Who are you going to contact to give those orders to? To the militia this looks like a suicide mission (and it might be), why should they follow those orders?

A large aspect of military training is to get people accustom to following orders and to recognize and accept their chain of command. By nature most people question what they are told, procrastinate, or otherwise avoid doing unpleasant things (like letting someone shoot at them). One of the primary objectives of basic training is to break down this natural inclination and get people to do what they are told, and do it right now! In combat there is often no time for question or explanation.

You need trust between the different layers of the organization. Front line troops need to react when given an order because delay means the situation has changed and what would have helped is now going to hurt or be pointless instead. They also need to trust that their commanders have information that they do not, and are not going to throw their lives away frivolously. Your lower and middle tier officers need to be able to read the situation and know when their orders no longer apply, when they have information that the upper ranks do not, when the situation allows for them to question what is coming from HQ, and when they need to act without waiting for orders. Your high command needs to trust that their field officers know what they are doing, and have quicker access to local information than HQ does. They need to judge what information needs to reach different areas and how to best use the troops available.

A few of the other answers imply that the civilian police forces can step in to provide this command structure. But that is not how they are organized or what they are trained to do. Generally speaking, police forces are local organizations with somewhat tenuous connections to higher authority. The police in a small town know their populace and area, they likely know how to reach the regional HQ/directors and their immediate neighbors, but they most likely do not take orders from them or communicate on a daily basis. Similar for large cities, precinct A has a specific area that they are focused on and don't pay much attention to what is happening in the other areas. In general, police training and procedure are designed to handle momentary situations on an individual or small group basis, not ongoing battles and confrontations involving hundreds to thousands of people.

Geography

The nature of the terrain will also play a large part in the success of the defense. By putting your defenders on a relatively small island you have already hampered them, they just don't have a lot of room to work with. However, if there are significant terrain features that can be leveraged as defensive points or barricades it will help a lot.

Dropping the bridges across major rivers will do a lot to slow the enemy. Mining them so that they can be blown mid crossing has the potential to do even better, but adds the risk that they will be able to disarm the charges or otherwise prevent you from detonating them, thereby securing a crossing.

Significant mountains implies passes that can be barricaded and defended, and opens the potential for things like triggered rock slides or avalanches to take out enemy columns or make the roads impassible.

Lots of heavy forest or swamps give places where your people can hide and launch harassing attacks while making it difficult for the enemy to maneuver.

Weather & climate are also a potential factor. The locals are accustom to it, they know which direction storms come from, how fast they move, and how severe they tend to be. They know what temperatures to expect, how they change through out the day, and how to live/work with them. If the invading force is not familiar with these things it is going to hamper them. If they are from significantly warmer/cooler climates they are also likely to encounter weather related injuries such as heat stroke or frostbite.

Economic

The nature of your countries economy will also play a part. Do they have significant natural resources and the infrastructure to turn them into finished products? Are such resources and infrastructure spread out enough to prevent the enemy from seizing them right off? Can the things that are produced and associated infrastructure be repurposed for war? Can such repurposing happen fast enough to make a difference?

As an example, if the country has a significant mining industry, that implies a fair amount of heavy equipment and raw materials to work with. There is also a decent chance that there is a ready supply of industrial explosives around. In the end, explosives are explosives, they can be used to build various types of mines, collapse bridges, destroy roads, and create various improvised weapons. Heavy equipment can be used to modify terrain, either in building defensive positions, or raising other obstructions. It is also conceivable that some could have armor bolted on and be used as APCs or other fighting vehicles.

Another key economic issue is whether the country is a net importer or exporter of various goods. How self sufficient are they? If the enemy just seized their primary port, how long are they going to be able to keep power on? run vehicles? feed their people?

Finally what is the lifestyle of the general populace and what are they accustom to? Societies which are accustom to lots of creature comforts and ready access to food, power, and other modern conveniences & luxuries are not going to fair as well when suddenly confronted with the hardships of war.

Modern America is a prime example, generally speaking, people are accustomed to being able to walk into any of a dozen stores on any day/time and being able to find what they need, be it a loaf of bread, clothing, a new car, or a high powered rifle. If something were to cause a disruption in the supplies or power flowing into a city, most of the people there would have no idea how to deal with it. More than a few days of disruption and society would start to collapse as supplies run out and people start to turn on each other.

You're missing the big picture

A classic Army for force-on-force enagagements must be trained and equipped before the fight. That takes time and money. Your scenario indicates that's already too late since the fight is already in progress.

At the National level, before fleeing the capital, leaders must decide upon a strategy to dislodge the invaders: Should they solicit foreign help (Kuwait), or expel the invaders without assistance (Finland)? The types of organizations, targets, equipment, and responses will be based upon the chosen strategy. For example, the strategy will determine the main effort of any government-in-exile: propaganda vs. fundraising and logistics.

Lower-level national leaders and local leaders form the command and logistical framework necessary to bind the many soon-to-spring-up independent resistance cells together into a unified national effort. This takes a bit of persuasion, particularly if local cell leaders disagree with the national strategy.

For example: In a foreign-assistance (Kuwait) strategy, expending resources to disable invaders' air-defense (at the right time) is sensible. In a do-it-yourself (Finland) strategy, that would seem a waste. If your cells are uncoordinated, then air defenses might get attacked, or not, or at the wrong time. Lots of wasted effort and lives that way.

The national command and logistical framework provides unity of effort toward the chosen strategy, prevents destructive competition between cells, and maximizes the effect of limited resources.

There's more. All a resistance or insurgency can do sap the invaders' initiative; to transition them from offense to defense. Historically, guerilla forces cannot expel the invader - they are not trained or equipped for that kind of fight. So once you have denied the invaders mobility, start building your Army in an obscure corner of the country. A real army with a general and uniforms and perhaps some tanks.

There's a reason for this: The new army clearly, unambiguously, and legally represents the old government. It's a legitimate force that can be negotiated with or (eventually) surrendered to. Even if it's just a battalion. It's a rallying symbol for the occupied population. If the invaders turn over territory to a local quisling, the presence of the army makes it clear who is on whose side. If a foreign power intervenes, the army is the organization that can be coordinated with, and liberated territory turned over to.

They don't need to mobilize an army?

Even without the military, the US would be a hard country to invade. The general population is nationalistic and there is easy access to guns and ammunition. There is also a number of gun enthusiasts with reasonable training that could manage and train smaller fractions of resistance. This is hard to fight against as an invading force...

Iceland has a population of around 350 thousand people. If slightly more than 10% decided to try and sabotage the invading army, you'd get twice the number of guerrilla soldiers as the invading army. If they're nationalistic or the invading force is cruel, they can get significantly more. That's going to put a damper on the invasion real quick.

They might not have a standing army, but there is nothing to say they can't have resources available. Emergency stores with supplies such as medicine, food and water could go a long way. If there's some weapons and ammunition available, that would be great. For that sake, there is nothing to say that the country itself isn't a big exporter of weapons.

Secondary question: Yes they can ask whoever they want. If they will get help is another question.

Twitter.

Ok, that's simplistic, but consider how the Arab Spring played out just a few years ago. The details are different (not an insta-army in response to an invasion), but the principle applies: in a place with nationalistic citizens, guns, and the Internet, loosely-organized resistance is possible.

A real army requires years of preparation, but army-on-army isn't your only possible response. You would be better off facilitating guerilla warfare. You're in for a lot of chaos; you won't be able to centrally-organize this stuff. But if the cause matters enough to your people, you can encourage them to take matters into their own hands. You'll probably have to make it known that you will look the other way about how they slow down the invaders, and you'll have some work to clean up afterwards, but you might not be doomed.

You can!
If you can't handle the problem yourself try making it a problem for other powerful countries. You can come to an agreement by trading your valuable resources to another country with powerful army forces at a cheap price (eg:- oil, gold). So you can expect them to fight for you.
Remember that making allies is an important strategy in any warfare. With the right resources at stake, you can defend/defeat any force. The resource you choose to trade doesn't necessarily have to be gold or oil any other kind of trade item. Even sometimes it could be the country's geographical location that carries an immense economic advantage.

They don't have to have an army. As world history shows to us, guerrilla partisans can resist to invading forces quite a long time even in very small country. Especially if invaders are not that familiar with the landscape environment, local culture and all this staff.

Look at mid-Asia countries as an example: how much decades they're hot-points in terms of warfare combat, attempts to re-format their culture to alien ideas etc? And regardless the amount of blood spilled and tremendous budgets spent, any invasion status is sub-zero success. Any woman can be a kamikaze bomber, any errand-boy can stab you with a knife suddenly, any elder can be guerilla informer.

You mention that citizens are allowed to have weapons, but few do. However, the fact that they can means there's a supplier of firearms. If everyone in the capital purchases a weapon and the police force starts organizing a paramilitary style army for two weeks, they could at least put up a fight. If they demolish the roads leading to the capital, set traps everywhere and engage in extended urban warfare they could last a couple of weeks at most. Maybe that's enough for reinforcements from other allied nations to arrive?

You'll find your best guidance in history. The standing army is a relatively modern construct. Historically the greater portion of fighting was generally done by mercenaries.

Modern Mercenaries

https://www.businessinsider.com/bi-mercenary-armies-2012-2?r=US&IR=T

Arrière Ban

There's some great info in here:
https://www.reddit.com/r/AskHistorians/comments/24d1tz/mechanism_of_raising_an_army_in_medieval_europe/

Basically the option facing land owners in 14th century France was either to pay money to contribute to the recruitment of mercenaries, or contribute an amount of men. The second of these options was of course deeply unpopular, and reminiscent of modern conscription.

Modern Examples

The following countries have no armed forces:
https://en.wikipedia.org/wiki/List_of_countries_without_armed_forces

For many of these, they are small nations for which their international partnerships are very important. See Monaco as an example.

Asymmetric warfare, as others have stated, is definitely the key here. I wont belabor the main gist of that but would add a few things not yet addressed.

1. Owning or shaping the internal and external message are the real long term keys to success. It doesnt matter if you fail in every operation as long as your messaging resonates at home and abroad. Your three audiences are the internal population for morale and support purposes; sympathetic nations who may be swayed to pressure the invaders or assist you; the population of the invading country as they are footing the bill in blood and cash for this endeavor. Crafting appropriate messages and hammering them home across the spectrum of communication will likely prove more decisive than any kinetic action although kinetic action must be part of the mix to show resolve. Many examples of this exist but Id point to two classic historical cases (as in the conflict is completely resolved) which are the French and US involvement in Vietnam and the Soviet invasion of Afghanistan.




2. Co-opting the invaders from the beginning allows for long term success. It is possible to be completely invaded/subsumed and still win in the end. The invaders will not have unlimited money and will look to create efficiencies. If the police force is postured to appear happy to collaborate and the mid and low level bureaucrats present in the same way many will be incorporated into the operation of the new government and can set about slowly nudging the government into doing things advantageous to the invaded nation. It would be important to have strong nationalistic ties and craft guidance for everyone that is easy to follow in a vacuum. Some type of low key cell structure would help keep such an initiative on the tracks. Examples would possibly be former colonies that gained independence from their European "patrons" through non violent means such as India and Pakistan.

One final big question though. Why would anyone think the effort would be worth the cost to invade a resource-poor isolated nation? Some sort of geographic strategic value?

Your country does not have a standing army. But it might have a militia.

This was the way the US was originally intended to work. The writes of the Constitution were nervous about standing armies, and rightly so; they are dangerous in several different ways. But you need to be practical about the possibility of invasion or insurrection. A standing army implies persons whose entire job is to be in the army. A militia is like the National Guard - civilians with regular jobs who come together to train and practice, and who are available to mobilize on short notice when there is need.

From the US Constitution - Article 1 section 8, Powers of the Congress.
http://constitutionus.com/#a2s2c1

15: To provide for calling forth the Militia to execute the Laws of
the Union, suppress Insurrections and repel Invasions;

16: To provide for organizing, arming, and disciplining, the Militia,
and for governing such Part of them as may be employed in the Service
of the United States, reserving to the States respectively, the
Appointment of the Officers, and the Authority of training the Militia
according to the discipline prescribed by Congress;

Your scenario says no standing army. Maybe the militia is how they planned to do it. The citizen soldiers will be mustered, issued weapons and then do it like they practiced.

Bonus: if you are going to muster, you will get to have a Mustermaster.

Look at the Fyrd system from English history

This system was most famously implemented and perfected under Alfred the Great. Basically, each region is responsible for contributing military forces called up from the local population to go to fight. They are expected to be able to mobilize quickly and answer the call to arms, but much of the time, they are working in their own communities. This system was largely responsible for his success against the Vikings, and causing them to look elsewhere for raiding targets.

While the answers talking about militias are good, you undercut that possibility with this statement:

"Firearm legislation allows for all citizens to be able to own handguns as well as shotguns and rifles, but few do."

If there are generally no arms in the country, making them from scratch is doable, but takes TIME (not a couple of weeks for the necessary quantity). Ammunition is even harder to start from scratch (gunpowder takes a bit of effort).

More importantly than the manufacture of weapons (or capture - molotov cocktails are cheap) is a culture of arms and independence. If the society allows weapons, but there is no culture of using them, any militia/guerrilla army raised will be largely incompetent.
(EDIT: And Rozwell actually notes this point.)

Aid from elsewhere is possible, but will not happen in two weeks for a physically isolated country (unless there is something already arranged - which you say is not the case - and the invasion was signaled so that the other country could put its forces on alert).

"How can a country with no standing army for the last one hundred years mobilize an army?"

It depends on what you call an army. If you mean a human organization able to win over another army. It's possible under certain conditions summarised by: compensate enemy advantages with your advantages.

You have the numbers but that doesn't go long way. The most critical aspect of an army is it's morale which you can achieve without having an army. The morale depends of many things but I'll try to simplify them in three: th belief that victory is achievable, belief that the cause for defense is just (more just than your ennemie's one), love for your way of life (nationalism, solidarity).

In that sense the first and only way to win this battle is to first win the morale battle. Publicity that confirms these beliefs is necessary, while international efforts to descredit your enemy can achieve it too.
Conversely, avoiding information that would undermine these beliefs is equaly important.

Under that condition a partisan army will form itself with little exterior help faster than the government forces could.
You have all the weapons you need. You have some guns but you certainly have explosives to even take care of the armoured vehicles.

"Additional Question - Would the defending country's government be able to appeal to the UN or NATO for support/an expeditionary force?"

If your ennemy is Russia, fighting in the middle of the atlantic and with so little troops I would say they wont resist the opportunity to hit their foremost political enemy even if you don't ask them.

A modern military needs a lot of stuff, coordination, and support.

The enemy is invading an island (a difficult thing to do already) with only 500 support personnel for almost 100 vehicles and 20,000 line troops? That's a bold move. Most of their tanks and many other vehicles are likely to be broken down on the side of the road fairly shortly. Their troops won't be able to scavenge ammo from the locals because even if they find any it's likely to be a different caliber. Their medics and other support personnel will be too overloaded and won't be able to handle everything. They'll be sick, hungry, and in unfamiliar hostile territory.

Meanwhile, their ships are sitting ducks. So find an ally with a submarine or buy one and get enough people aboard who can figure out how to operate it just well enough to sink the enemy supply ships. Then the locals can round up the remains of the enemy troops with pitchforks.

Nope.

Look at what happened in Yugoslavia in WWII. Individual partisans with high morale hastily formed into groups consistently lost to poorly-armed, poorly-led and poorly-equipped regular troops. You might have enough time to get much of the population familiar with their weapons. Turning them into actual military units will take much longer.

Guerrilla warfare can take place and make the place difficult to govern, depending on the motivation of the defenders and the scruples of the attackers. It was a high-cost operation in the German Eastern Front areas in WWII.