Actually Scottish independence is reasonably likely in the next five or ten years (not probable, but double-figure-percentage likely).
The independence party (Scottish National Party) have said that they would remain members of the EU and would not enforce any border restrictions, so it wouldn't be much different to the existing land border between Northern Ireland (UK) and the Republic of Ireland (hasn't been UK since, erm, 1920 something?), where there are already lots of roads and railways crossing the border with no border checks whatsoever - just fire up Google Maps Street View and you can follow roads betwen NI and ROI with not a border check in sight (most of the time there isn't even a road sign telling you which country you're in). Being a member of the EU would also mean that Scots would continue to be able to work legally in England, and Englishmen would continue to be able to work legally in Scotland.
The SNP have also said that Scotland would remain a member of NATO so the use of the Faslane nuclear submarine base would be pretty much unchanged. The only real problems would be beancounter stuff like exactly who owns the submarines, warships, fighter jets and tanks.
There would be similar beancounter problems in genuinely trans-national government-owned entities such as the BBC; for instance the BBC's children's programmes are mostly made at dedicated studios in Scotland, whereas the BBC drama programmes are mostly made at dedicated studios in the English Midlands and BBC sci-fi programmes are mostly made at dedicated studios in Wales. But for most entities, including the socialist-model NHS, it is already broken down into regional or sub-national components anyway.
The UK is pretty much already set up for independence. Since we're already in the European Union it really wouldn't make much difference.
Even the Queen-as-head-of-state thing is already covered by the Commonwealth. There are lots of independent countries which have kept the Queen as head of state; Canada, for example.
A lot of English people are actually quite pro-independence for Scotland, since Scotland has a poorer economy which gets subsidised by England (although you could argue that most of the UK is actually subsidised by Greater London).
It might happen, it could easily happen, there aren't any big practical barriers, there'd be very little day-to-day change and most British people are pretty indifferent about it. Scottish independence? "Meh".
(Disclaimer: I'm English living in England)