Not so sure, WLD. Back in the 80s, the Cold War was still very much active. It was only 10 years since Vietnam; Reagan was doing his best to provoke the USSR with his SDI nonsense; and the main fear was still the Russians coming over the plains of Europe.
Things are a bit different now though. Nuclear weapons simply won't deter terrorists/freedom-fighters, and their use in areas which inevitably have huge civilian populations would be madness. Even if some nutter uses WMDs (most likely chemical or biological), I think the UK people simply wouldn't stand for nuking the entire country. I hope the US is the same.
The priority now seems to be troops on the ground with technological superiority, and neither the US nor the UK can afford vast nuclear arsenals *and* lots of troops on the ground *and* schools/hospitals/other public services. The cost of Trident would get you an awful lot of regular cruise missiles or trained soldiers (or conversely some serious improvements in schools and hospitals, but that ain't likely).