If it comes to that, the current global society is done and gone, and quite possibly humanity with it. No way the UK doesn't have control over their own devices though.
France built an independent system, the UK did not.
https://www.gov.uk/government/publications/uk-nuclear-deterr...
"Although the UK’s nuclear deterrent is assigned to the defence of NATO, we retain full operational control over its use. Only the UK Prime Minister can authorise the use of our nuclear weapons, even if used as part of a wider NATO response."
It's also just not at all believable that they would not be able to push their own buttons. Why would it be set up that way? And what would prevent fixing it even if it were?