Not been around for a while but this series of events make me want to rant a while. To avoid this have some amateur legal and political speculation as to what might happen next. I should go back an edit in some links but have wasted enough time on this for now.
This is broadly what would expected based on a leave vote if the vote itself was legally binding (it wasn't). The process is broadly as follows;
1) A notification is issued by the Prime Minister to the European Commission.
This step in itself is complicated politically and legally speaking. Under current constitutional convention it would require the assent of the devolved legislatures of Scotland, Wales and Northern Ireland, but that could be overruled by Parliament as an exercise of sovereignty. How the union survives that is unclear. Whether it requires the assent of Parliament is less clear. The legal experts I follow seem split on the question, although it is less clear to me that the Prime Minister could unilaterally ignore the devolved Parliaments.
Had the referendum been binding then this is the step that would (or should) have been made automatic and there would have been no problem with that, Parliament being Sovereign and all that.
Anyway, let's assume that a notification is made, it stands legally and the EC accept it.
2) Britain negotiates a withdrawal agreement.
This is where the 2 year number comes from. This number is very soft. At any time a qualified majority (20 states) can sign an agreement with the assent of the European Parliament, ending negotiations. Similarly the time can be extended, although that would require all 28 parties to agree.
During this period Britain would no longer have a seat at the table but would still be bound by European law, although the extent of this is debatable.
There is split belief as to whether we can unilaterally stop the process at this stage. Article 50 doesn't make explicit allowance and most experts I have seen suggest we would have to have unanimous consent of all 27 other member states to rejoin even if negotiations haven't ended yet. Changing our mind would require everyone else accepting that.
These negotiations would be relatively narrow, mostly deciding political points that are required for a withdrawal to be possible. They would not cover trade. Rather they would be issues such as how to treat EU/UK nations living in the other area and how to split assets common to both bodies. How these issues are handled if negotiations end without an agreement (either due to the two years expiring or them being terminated early) isn't clear to me and I haven't seen any discussion of the option being taken seriously.
3) Britain is out.
At this stage we are out of any treaties we signed as part of the EU, including every trade deal we have. We would be under WTO rules for any and all trade we want to continue with until new deals can be arranged. The likely first such deal would be with the EU itself.
The nature of this deal is open for question. The first would be a role within the EEA under the Norwegian model. Given that immediately after an exit our laws would still incorporate European law this would be a relatively easy deal to get, but it would require we retain much of that law, as well as being re-bound by many of those treaties we exited. In particular freedom of movement is seen as a red line by most EU nations. Even for a looser connection like that Switzerland has would be hard to get without binding ourselves somewhat more closely than has been suggested by the Leave side.
It is hard for me to imagine a deal that has us in the common market through the EEA or even just EFTA which is better than our current EU role, with our opt-outs and vetoes. But for business it may well be the second best option.
Going completely outside the common market would make it far harder to gain many of the benefits we currently take for granted and use for leverage in other trade deals. Particularly the benefits our banking and service sectors enjoy are pretty much non-existent outside the common market. However, it may well be possible to scrape out a deal along those lines. This is the only option that removes freedom of movement (IMO).
Only once we have negotiated a deal with Europe (and we may be several years past exit at this point - 3 to 5 has been suggested) are we likely to be able to negotiate deals elsewhere. Our trade relationships with Europe will be dominant and dictate how attractive a trade partner we are. Would we still be a gateway to Europe for nations like the USA, India and China? Or should they look towards Germany and Ireland instead? Those trade deals will then take longer to negotiate. China have suggested 10 years while 5 or so wouldn't be surprising for a substantive bilateral trade agreement. How many such deals Whitehall can work on in parallel is an open question, but it's unlikely that any British diplomat will have any free time for a good couple of decades before we replicate the wealth of agreements we currently enjoy through the EU.
We could also look to be included in large multilateral agreements, but that would be as a junior compared to the giants of the EU and USA. If treaties like TTIP look bad imagine them when we don't have the clout to get even halfway favourable terms.
As far as European law goes, we could begin the process of removing it. Of course, no-one is quite sure how or what would be removed. The European Communities Act of 1972 incorporates something like 60 years of EU law into British law, and the process of repealing it is unclear. For sure it is too large a project for Parliament to debate and decide on each individual piece of legislation if they plan on passing any other law or doing any campaigning in the next ten years.
Repealing the Human Rights Act (and so withdrawing from the ECHR) would also be an option, but the Conservatives have been trying to write a British Bill of Rights for years now and haven't gotten past the title, so what such a repeal would look like is unclear.
At this stage EU funding to areas of the UK currently enjoying it would be lost. This particularly hurts Wales to the tune of £500 million, but also Cornwall and a number of deprived cities and regions. While Britain pays in more than it receives by the EU, that doesn't mean we can immediately re-distribute these funds. Regional funding in the UK is carried out under the Barnett Formula. Changing Barnett is politically fraught, mostly as it is very favorable to Scotland and its continuation was pledged after the In vote in their independence referendum. A London government getting in the business of controlling regional funding again would be painful for any party and possibly the last straw that triggers another independence fight (if that doesn't happen anyway).
Similarly farmers would lose their funding under the CAP - around £3 billion a year; ~55% of farmer's income. Britain could, if no longer paying into CAP (we pay £6bn a year), afford to match that funding easily but then would be in the business of directly subsidising farmers. Given our governments have been deftly blaming the EU for farmer's woes on the EU - despite DEFRA actually being the enforcement/funding body - that could again hurt any government politically if the distribution is not to farmer's liking, or if free market types complain about government handouts propping up dying industries. Similarly the fishing quotas (required so we don't fish stocks out of existence) would have to come from British politicians, not the convenient faceless Brussels bureaucrats.
To make life more hellish, the Good Friday Agreement is conditional on our and Ireland's EU membership. We could probably salvage the deal by putting in place replacement structures, but the solution would not be simple or quick. Messing with the Northern Irish peace process will not be welcome. Similarly we would have to revisit the question of the Irish border and citizenship. Some of this is simpler if we remain in the EEA, retain the ECHR and freedom of movement, but the more things change the more we would have to do to solve the Irish question again.
Basically, no British politicians really want to go through this process. It would be messy, long, costly economically and likely fatal politically. I'd say the most in favour would be Farrage, as there is no chance of him actually having to do any of the work or taking any of the blame. But as a rule most in government, politics, business and life in general are looking at the complexities and potential pitfalls and noping out.
Under point 1 of the first option I mentioned we have to notify the EU of our decision to leave. This has to come from the UK; the EU can't force the process. If the government decides not to issue a notification then we never have to take any notice of the referendum. Of course, this might not be politically popular.
Instead there have been suggestions, mostly from Leave in the last week of the campaign, that we might start passing laws as though we had already left. This would likely leave us in breach of European law, allowing the European Parliament to invoke Article 7 of the Treaty of Lisbon. This article allows them to suspend our rights and representation. While there is no facility to expel a nation, once the UK is removed from voting a new article or amendment could be passed allowing for our expulsion. This would take all 27 other states agreeing to it, so it is a remote possibility, but would let British politicians off the hook and allow them to blame any economic damage on the EU pushing us out before we were ready.
This would then skip directly to stage 3 of the first process, although likely in a far weaker negotiating position as we have just proved ourselves willing to ignore treaty obligations when it is politically convenient.
I only include this because it is the only way in which the Leave campaigns proposals to unilaterally ignore European law makes any damned sense. Much more likely is they will ignore those proposals along with all their campaign promises.
Boris Johnson proposed last year that we hold a referendum, vote leave, then re-negotiate our position, hold another referendum and vote remain. This option is currently being dismissed by absolutely everyone but is easily still open. After all, absent any treaty violations or an Article 50 notification we can't be removed from the EU and so basically can continue in the current state indefinitely. Refusal to hold unofficial negotiations can only last so long.
We now have 4 months of uncertainty till there is a new PM and the withdrawal process can begin. That is 4 months of uncertainty and pain for businesses. Job losses and hiring freezes, budget cuts, contracts abandoned, etc, etc, etc. The prospect of more could sway people away from going through with a withdrawal. Even relatively minor concessions from the EU (who would also be going through similar pain and calls for reform) might be enough to allow the new PM to trigger a referendum and remain vote. Or even just continue to ignore the referendum if they are willing to eat the political cost.
This would require a change of tune from the EU in the relatively near future, but I still see it as potentially the most likely scenario. I don't see any appetite for withdrawal from any of the likely Conservative leaders and I feel the option will look more and more likely over time, especially if we end up with Boris as PM.
Of course this would make us horribly unpopular in Europe. We would be talking about a year (at least) of horrible economic damage for what is basically an internal party leadership battle. It would also be unlikely to appease the harder line UKIP/Leave voters, keeping the prospect of further exit campaigns open.
Parliament in the UK is sovereign. That means they don't have to care about anything, they do what they want and what they do is reality. They could completely ignore the referendum. They could do anything.
Of course, that is ignoring a political mandate, which is usually a bad idea. However, they could always get a new one. If there was a new general referendum held and it was won by a party campaigning to stay in the EU then they could quite safely ignore the leave vote, at least for the duration of the next Parliament. The Lib Dems have already said they will fight the next election on a platform of being part of the EU, whether that means remain or re-entry. Labour are currently self destructing and could go either way on this issue. Mostly this depends on the Conservative leadership battle and whether a general election is triggered before an Article 50 notification happens or not.
This would almost certainly be the messiest for internal UK politics and would require a serious collapse on the right along with a resurgence on the left. Until Labour rebuild their leadership and the Conservative leadership battle shapes up it isn't clear if this is anything beyond a hypothetical to explode Leaver heads.