Independence is a big word, Sovereignty is perhaps a bit better.
If they would achieve that, they would be outside the EU as well, because GB, Spain, France an a few others would veto it.
They are too afraid it would encourage their own Nationalist Basques, Catalans, Corse an so on.
It would be in the UK's interest (NB: UK, not GB) for an independent Scotland to be seen as a "continuing" state rather than a "successor" state -- successor states are new countries and do not inherit debt from the parent country. As the UK would want Scotland to take a proportionate share of the debt, the UK cannot veto Scotland's membership of the EU, of NATO or of the UN.
France or Belgium might object, but Spain's stance really depends on who's in power at the time -- Rajoy has made noises against Scotland, but some of the smarter heads in Spanish politics and diplomacy have hedged their bets more, going with the Yes Scotland line that Scotland is a completely different case due to the Treaty of the Union and the Edinburgh Agreement. I say "smarter heads", because while Rajoy doesn't want Scotland to become independent and set a precedent for Catalunya to follow, these guys appreciate that accepting the Yes Scotland line means that whatever Scotland does, no precedent can or would be set.