Odds are increased that Scotland will eventually go for independence. Northern Ireland is closer than ever to a Republican majority in Stormont and speculations are on the rise that a border poll might be on the table in the near future (as in within not too many years; it will still take time) and even some unionists appears to accept that there is a greater than ever chance of a unified Ireland. More importantly: at least some unionists appears more accepting of that possibility, as they see Brexit as yet another example of Norther Irelands concerns being ignored - how the negotiations unfolds will potentially have a dramatic effect on the Northern Ireland issue as a result.
Irrespective of that, it's pretty clear that Brexit will force Westminster into promising drastically increased levels of devolution not just to Wales, Scotland and Northern Ireland, but also to the major urban regions like Greater Manchester and Greater London that will reduce the de facto power of Westminster dramatically and allow the regions to drift even more apart in terms of differences in public policy.
At the same time there is a major split amongst those who considers themselves predominantly or only British or who considers themselves British first vs. those who predominantly see themselves as English, Welsh, Scottish or Irish first, and British second or not at all, with "Britishness" being weak and fractured.
Basically, to a lot of people living here, the UK means as little or even less than the EU, compared to other parts of their identity.
And a lot of the political discourse has been around how the elites ignore people. What people are about to find out is that in that respect the London elite were the ones ignoring them, not Brussels, and accordingly Brexit will not solve any of the problems they had in that respect.
Expect to see increasing upheaval and pressure for more power to be devolved from Westminster as a result. E.g. after the Brexit vote there was a brief flash of people talking about pushing for independence for London, and while that won't happen anytime soon if ever, expect to see those debates flash up again and lead to an actual (if miniscule) political movement at some point over the next couple of years.
I personally think that while the UK may persist in some form, the UK 10 years from now is reasonably likely to exclude Scotland, and the UK 30 years from now is likely to exclude Northern Ireland, and be more like a loose federation with massively devolved powers to Wales and several English regions and at least Manchester and London. Maybe even with increasing support for Welsh independence.