The same happens as always happens when countries divide and borders change. It happens all the time.
I think maybe ropedope (and I) are not as familiar as you with what happens "all the time" - maybe you could enlighten us as to what the likely outcome would be.
I guess that the contract for the Scottish bases would terminate, and that government of the newly independent Scotland would decide on their own requirements and go out to tender, but I know very little about these things, as I have not seen them happen all the time.