I'm curious. Would this change anything to the world situation?
Even without the butterfly effect then yeah it would probably send out some decent ripples, though it would not be as world changing as Russia and China teaming up or something.
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It would however change depending upon what happened next.
Three scenarios. In no case do I expect it to become a military dictatorship or anything truly silly. The UK would likely still be a member of the UN and Nato and all sorts of other agreements/groups like that, I am not sure what might change there (the UK is a far bigger player in some of those than the population and military potency might suggest and in some cases it is so as a kind of EU proxy but not really). No country is likely to go bankrupt instantly because of it and nobody is likely to see a massive boon either.
The UK become a somewhat isolationist type country a la Russia of 10 years ago (or indeed Russia prior to this latest round of nonsense), whether by choice or by circumstance/"punishment" I do not know. You then need visas (or maybe visa waiver is continued), immigration is a pain and emigration is the same as people coming from places that aren't white face today (maybe think those moving to the US -- family, money or serious education* in areas somewhere else wants). Trade agreements exist but nowhere near as free and easy as they are today, possibly as a certain penalty for leaving from various other places (France and the US already threatened such things, Germany on the other hand said it would be nice, possibly because it sells a lot of cars to the UK market). Seemingly a wet dream for certain political parties and would be a nightmare for others.
The UK leaves the EU but stays within the EEC/EEA/other things that could technically count as Europe (which are all separate, if they technically still exist). Not sure what I want to point to as a historical example, mainly as the choices for similar models are far smaller or were heading towards EU membership anyway. The extent of this is a bit of sliding scale but who knows. I think this is what most of those campaigning to leave hope will happen but seemingly have not got in place.
It stays, business as usual. Some might remember, most would forget fairly quickly and probably even if the results are tight.
*it is worth remembering that emigration is a major thing for people graduating from UK universities -- science, engineering, maths, medicine, IT and such all see lots of people leave after things have finished or once they have a bit of experience. The EU is a significant component of that. Said same fields also often retire to various places in Europe where I am told a lot of places then get a significant income. If it was then as much of a ball ache as moving to the US is it might change what people go in for as the UK would not likely be able to sustain as many graduates as it makes in the lifestyle to which they wish to become accustomed (IT is rapidly falling and science and engineering in the UK is not a lucrative field for the amount of time you need to sink into it to be able to do it).
Some have pondered what might happen to the other non England countries in the UK. The results of the last election are that Scotland is largely run by the SNP (a political party primarily concerned with Scotland, what with it being in the title), this comes after a very narrow referendum within Scotland on whether to stay in the UK. Depending upon the voter breakdown it might see a call for another referendum, though curiously the EU was a bit hazy on what position Scotland might take during the independence referendum so that might be interesting. I doubt anything major would happen to Wales or Northern Ireland, though it is not impossible that something could happen to Northern Ireland with regards to joining the republic or strengthening ties there. You also have things like Gibraltar (a city sized area on the south of Spain that the UK sort of owns) but that gets complex and we would also have to consider the rest of the former empire, several of which are variously renegotiating or otherwise contemplating their level of ties with the UK and not having the easier access to Europe would factor into that. I doubt if the UK stays and voter lines in Scotland say leave that it would make for another referendum there but the other way around might accelerate things towards another.
It is worth considering also that nobody has ever left the EU, much less a major player within it. For all I mentioned UK internal politics then various other parts of the EU, including said major players, also have their own internal matters to consider and the UK blazing a trail might have knock on effects there. It is not unlike what you see in Quebec for some of those, though possibly with longer and more complex history (people see countries in the EU as old, they are not always and former city states and regions are prominent features today within lots of them). I do not know if it would be quite as huge as a state that matters leaving the US but it would pose similar issues.
The EU itself then. In terms of money in and money out directly (if we are doing secondary effects then we will be here all year and tertiary is too complex to even consider for me) from taxes and whatnot then the EU gets more from than it gives to the UK. Though it is nowhere near enough to massively destabilise things if the UK did stop contributing, though at the same time it would probably see budgets have to be redrawn for reasons other than the UK no longer getting things (I am not sure what things are in agreement to be funded for so many years regardless). The EU does fund a lot of building works and arts and sports and education and other public works that governments also attempt to fund within the UK.
Equally the UK is probably not a major draw for the EU as a concept -- nobody really sits there and thinks "ooh if we have good relations with the EU then we have de facto access to the UK". It might be a perk for some and a reason to go for it but the EU - the UK is a potent entity still. At the same time if the EU can not hold it together and keep the UK in it then it is not going to be a point in its favour. I don't think anywhere would suddenly start speaking to the EU if it was less the UK a la like some places pull out of things if some others decide to call Palestine a state.
For companies (individuals is a very different matter) the UK is not a tax haven for anything really (Monaco and for tech then the Republic of Ireland being the big ones there), nor a particularly useful place for lax laws for finance or environmental laws. It does have a fairly potent financial industry though which would see some things happen. To that end I don't imagine it will see things like what the Republic of Ireland saw a couple of years ago when they were mulling over things and even more attractive places for tech companies decided to pop up. It does however have the biggest stock exchange in terms of market cap in the EU by a considerable way, though volume wise the next European one is greater.
The UK does have a lot of very nice ports*, several of which are largely owned by China and deal with Chinese ships an awful lot, several of which are truly ridiculously huge (post-panamax if you want a term to go searching with) and the EU does not have many things that could otherwise handle them, Germany being notably absent from the list (though with countries on all sites having them and being close to their major cities it is not a big thing). Despite then having to be shipped out afterwards to the mainland the UK is a gateway to Europe for a lot of things. If taxes and tariffs get complicated then the UK becomes a less useful destination and it is doubtful whether the domestic market is big enough to otherwise sustain them in the long term.
*more than any other country in Europe, and a significant percentage of the options in Europe.
I must confess I am not sure to what extent the UK influences EU policy on things. Those that encourage a vote to leave say they want Brussels (by which they mean the EU) to stop controlling UK laws which is a near criminal oversimplification of things as the UK does send people to oversee, debate and vote on what happens. Do also remember that the EU enjoys a somewhat better relationship on the laws front than the US Federal government does to its states. I could point to 10 examples of US states and federal government butting heads from my head, other than this whole migrant thing and maybe the handling of Greece I have not really got much like that for the EU (though said handling is probably the only reasons the referendum in the UK was able to get any real traction). At least for the EU to the UK, some of countries where religion is a bigger thing do occasionally have some issues with some things. Their one nation that is Europe thing they aim for is often poorly executed but seldom to the extent it truly bothers people and most treat it as something of a running joke. The UK potentially not answering to the EU court of human rights would also possibly change things a bit -- a lot of rulings it makes on tech and bioethics are fairly far reaching and not always what the top level UK courts could be trusted to do, though at the same time the ongoing EU vs Google spat showcases the other side of things there. Patents might also be interesting if the UK does not inherently be part of the European patents any more, though that would probably be a minor thing at best.
If it sounds like I mainly looked at internal UK and Eu politics it is because it is where most of it would play out -- the UK is a small island nation, one that acts a bit bigger than its population might dictate compared to elsewhere. It is however one that has very little in the way of natural resources, exportable agriculture, bulk industry or high tech industry you can not get anywhere else (they do exist but none are a major factor in the UK or in their respective markets), it is then what you might call a service economy. There would be ripples and certain things like if you ship something or want to have a bank account somewhere or possibly how it works if you want to go on holiday/travel for business.