On the subject of independent deterrence I believe much was made of RAF Thor IRBMs being under dual key control and only able to be used if the White House agreed. I also seem to remember at least one comment from a former Thor driver that if necessary the US-held key may well have been exchanged for a bullet from an RAF-issue personal weapon had the situation demanded it.
Don't expect the current arrangement vis-a-vis Trident to be quite the same, but I have no doubt that the UK can use it unilaterally if needed -warhead or no warhead
.