Sunk,
Again, it depends on what you think the "independent deterrent" is for. In the old world - an old world that might reconstitute itself someday - it was to deter against a calculated nuclear strike from the Soviet Union, one that was calibrated to be (in the eyes of a US president) not worth Atlanta or Portland.
That was why both the French and the Brits found it essential in the 1960s to consider a hedge against limited ABM developments (France's supersonic low-level CMs and the UK's Chevaline) which would have done little against a Soviet or US attack but would have rendered their independent ballistic missile forces useless.
So, where might a nuclear strike come from? And would it have an effective counterforce element? If not, Storm Shadows, Typhoons and tankers would survive. So VC (godless Communist and cat molester that he may be) could be right.
As for the need to maintain the nuclear submarine industrial base... it seems that, they way things are going, the UK is well on the way to dismantling its independent combat aircraft development infrastructure and becoming a subsidiary of Fort Worth. So which is more valuable? World it not make more sense to join forces on SSNs with the US? (While, perhaps, sounding them out on the future role of advanced non-nukes?)
The overall problem, increasingly, is that (a) CVF/JCA looks as if it is going to starve a lot of other efforts to death; and (b) in that light, is it important enough, or good enough? If we can't fix (a) then we have to answer (b).