Most EU countries have the luxury of not having to maintain a nuclear deterrent precisely because the UK and France maintain independent nuclear deterrents.
Not true. If nuclear deterrence works in a post-Cold War environment (big if), then the non-nuclear European states will be benefitting from the nucelar shield provided by the USA, not the piddling little arsenals of the French and the British.
Who, exactly is the UK's nuclear arsenal designed to deter? Obviously not the states with so many more weapons that the whole basis of nuclear deterrence (having a second strike capability, so that your enemy can't just wipe out your nuclear weapons in a sneak attack) is removed. So, our nuclear arsenal is no use against either Russia or China (or the USA if things got really weird). So who does that leave? Israel, India, Pakistan? None of them are going to launch a nuclear strike against the UK. North Korea appear to lack the capacity (so far) to produce long range nukes and in any case the UK would hardly be top of their hitlist.
Like the non-nuclear members of NATO, the UK is covered by the Article 5 guarantee in the NATO treaty which says that an attack on one member state is an attack on all. So like, say, Belgium, we benefit from the US's nuclear arsenal. It's this, rather than a handful of British missiles, that offers us and the other European NATO states protection from the kind of threats that might be deterred in a conventional way (e.g. threats from states not from IS). That's why Russia can invade Ukraine but not Finland - Finland is covered by Article 5 and therefore by the US's nuclear and immense conventional forces; Ukraine isn't but wanted to be, so Russia got in there before it could join NATO and have the same protection.