I think it is a bit stretch to consider the UK submarine forces or generally the armed forces as independent from the USA.You must know more than me, but AFAIK the UK has only built about 29 nuclear submarines to this day. I dont think that they are behind the US in terms of nuclear submarine tech.
The UK using USA made SLBMs, USA industrial persons helping the manufacturing of submarines, and the supply chain shared with the USA as well.
I think this is the case with the French program as well, the only french carrier using USA made catapults, so we can assume the submarine forces use foreign machines / equipment, and most likely there was external help regards of design/construction as well.
There are only three country making advanced military equipment independently at the moment, USA, Russia and China, and the later join late this club so playing catch up with the other two.