We do have precedent - Czechoslovakia is now 2 countries, as is Sudan and Yugoslavia.
I just hope Scottish independance goes better than the latter two.
Scotland may have some issue if they decide to get out of the nuclear game, mainly because of the number of jobs that will be lost when the base (and supporting businesses) gets closed.
At that point, Britain will almost certainly move the base to somewhere else in the UK, probably somewhere with lots of unemployed, ex-heavy industrial workers. As a result, they do have very good political value.
There is talk of just scrapping it completely, and relying on nuclear cruise missiles and bombers instead. I'm not sure how practical this is, but given the relatively stable state of nuclear politics (ie who would we potentially nuke today with our deterrent?) it doesn't seem so bad an option. The money could be funneled into traditional forces that end up being heavily used for peace keeping and humanitarian purposes.