Minus the apocalyptic soothsaying by both parties:

What basically has to happen now is the UK and the EU have to re-negotiate how they wish to do business. This is likely to both take a while and increase the standing of the UK in the EU. Meanwhile, it's looking like Scotland and Northern Ireland are also seeing this as a cue to attempt their referenda again, thereby miring the UK in particular and the EU in general in how, exactly, they choose to do business and conduct law.

It's possible/likely that other countries will make similar moves, thereby essentially forcing the EU to redefine the EU, which is going to take a while. It's going to make it a lot harder for the EU and its member states to trade in an agile fashion for a while.

The world will not end. Dogs and cats will not sleep together. HOWEVER a whole bunch of people who firmly believe they have better shit to do are now going to have to burn a shit ton of blood and treasure on treaties and trade agreements and boring shit like that while the rest of the world aren't.

Chances are good that whatever Europe looks like in ten years, it'll be ten years behind the rest of the world.

Make of that what you will.