It still undermines your argument.
It doesn't undermine the argument at all, because British political and military posture towards the Falklands in the early 80's suggested that they had no interest in defending the Falklands. This is why our conventional forces also failed to deter them, because they didn't believe we would act.
No one is saying that just by having nuclear weapons you'd impervious to invasion. There must also be a belief that you would use these weapons to defend your nation. This was not a realistic proposition for the Falklands at that time, or even today in fact. It was absolutely realistic if someone attacked the UK proper at that time.
Besides, the lack of invasion of modern western developed nations has more to do with globalisation & NATO than simply possessing nuclear weapons. Spain is part of NATO but lacks nuclear weapons, are they under the threat of conventional invasion?.
This is a gross oversimplification of the issue and utterly illogical. By your reasoning, Spain presently has no need of conventional forces because (as we agree) they do not presently face a conventional threat. But what if that threat did materialise in say five years time in a post-NATO world? You cannot expect to whip up a powerful conventional force overnight, so Spain would probably be overrun. The same applies to nuclear weapons, if you don't have them you cannot exactly whip them up overnight. If Russia decided to start to threaten us with nukes, we'd be defenceless.
Plus, there is no guarantee the US would even use its nuclear weapons to defend Spain, the UK or anyone else in NATO for that matter. Would you really put our national defence in the hands of
Washington?
The UK maintaining a strong nuclear capability means WE are able to protect ourselves. We are not reliant on our 'allies' and we cannot be bullied by our enemies.
Oh and its generally accepted that NATO would have been little more than a paper tiger without nuclear weapons, because the Soviets had a massive advantage in Europe in terms of conventional forces. It was the threat of nuclear weapons than guaranteed the Soviets (and Americans) would do everything they could to avoid an escalation.