Sorry mate, you have been reading too many warbooks. Our nuclear deterrent is what it says on the tin , ours. We are not beholden to NATO for its deployment. The scenario you post, confirms to me youknow little about the deterrent, and its raisen detre for the UK.
Our missiles are operated under NATO authority primarily. The public position is that PM has the final say in theory but has to "consult" with NATO before exercising the use of nuclear weapons. The MoD says there are circumstances where our national interests could be threatened outside the framework of NATO and that the PM has complete discretion in these circumstances.
So let's imagine a hypothetical scenario where that might apply. Argentina's in a mess economically (which isn't hypothetical) and there's civil unrest. The military are ordered to shoot protestors but refuse and instead mount a coup. To divert attention from their domestic issues they invade the Falklands. They know we'll struggle to mount a conventional response so think they've got the upper hand. But to make sure, they threaten to shoot 50 islanders a week until we cede the islands to them.
Our PM is a somewhat hawkish figure and threatens to retaliate if they do that so they call our bluff and shoot the first 50. The PM, knowing we are incapable of mounting a conventional operation, orders a nuclear strike on Buenos Aires. Do you seriously believe, regardless of anything the MoD might say publicly, that the USA would allow something like that?