Albeit in open chaos, the EU isn't going anywhere for at least a generation. It is too engrained in european politics and there are too many benefits for those with vested interests, i.e. the current paymasters. It would need to be simultaneously unpopular in Germany & France and that is just not the case.
As for the rest of the UK, I would say:
- Northern Ireland: Very relevant, Unionists would fear Scottish independence as the political split is now close to 50/50, Nationalists would welcome it. It is unclear whether an NI - Ireland unity would echo voting patterns until the Irish economy is back on it's feet. UK economy is faltering but basically pays for NI to exist. There is no real appetite in RoI just yet to swallow that burden. More of a mid term (20 to 30 years) project I would say.
Wales: Interested observers. The welsh parliament is the weakest of the 3 devolved assemblies with the least powers. Traditionally Wales is the least likely breakaway country from the UK union, reality is Wales is not strong economically, the do not have North Sea oil or a financial services industry / expertise as present in Scotland. Beautiful country, great rugby players, not ready to be a standalone entity financially.
England: I really don't know. A lot of English might well feel they are better off letting Scotland AND Northern Ireland disappear. I believe long term the burden of investment would see them significantly better off, they would need to accept a reduced role on the world stage as they would have a reduced military capacity and economy, but maybe that would be acceptable to a country tired of sending troops off to far off parts of the world? North Sea Oil and Gas does have a shelf life, it is a very mature field and not as vast as the other scandinavian fields.
No idea what the effect on our near Atlantic island neighbours in Port Stanley would be...mild nervousness. I imagine the Scottish ambassador would not want for social engagements in Buenos Aires....once they worked out how to open and pay for emabassies around the world !
Personally, from the heart I would like for Scottish and Irish independence to be viable, but the reality is that right now, neither are.