I think It's inevitable that eventually the United Kingdom will break up simply because it's served its purpose, it's run its course. The Acts of Union which brought it about happened at a time in history when they were politically expedient for those times. Times have changed. But it will happen gradually. Northern Ireland is not likely to reunite with the rest of Ireland any time soon. The peace there is much too fragile to risk it by giving Unionists the impression that they're being 'forced' into a united Ireland and the economy of the Republic of Ireland is going to be seriously impacted by Brexit, so the financial considerations are also an issue.
As for Scotland, they are practically independent already with their own parliament, legal system, education system, established Church and so on. They want to remain in the EU and are being forced out as a direct result of being a part of the UK so for them, the issue of independence is going to be to the forefront again and there may well be a second referendum within the next couple of years.
I think what's really interesting is the future of the British overseas territories and Crown dependencies like Gibraltar and the Channel Islands. They are not part of the UK but they will be affected by Brexit. Depending on which they were, they either got a vote on Brexit or they didn't. The Channel Islands didn't get a vote at all. Gibraltar did, and a staggering 95% voted to remain in the EU, yet they have to leave. It spells out very clearly that there is a price to be paid for being under the protection of the English Crown and enjoying the benefits it gives. The bottom line is that England is in the driving seat. Because the people of England voted to leave, everybody is out and some didn't even have a say in it. It's astonishing to think that people who consider themselves British, such as the Channel Islanders and the Manx people (Isle of Man) didn't even have a vote on this issue which will impact their lives so much. So I think there will be enormous repercussions in the long term.