"There is no way on this Earth we (as UK plc) could launch independently of the US."
Boll@cks! The Trident boats do not have US officers on board with a magic authorisation code like a Microsoft activation key. The UK could indeed launch independently if necessary. Whether from a political viewpoint we ever would is a totally different question.
A more relevant question is whether we need a nuclear capability that is based on a ballistic missile. Perhaps a move to lower yield cruise missile platforms may be more sensible. This would provide a much cheaper capability with the acceptable (?)disadvantage of a longer flight time.