The UK, France and the US are the only militaries in the world with any significant power projection - though ours is diminished at the moment due to a lack of air craft carriers. And the US is miles and away in front of the UK and France. But, put another way, basically no other country except the UK, US or France could have won a war like the Falklands. Though even then we had to piggy back on the US a lot even just to get to Iraq.
As for Trident, I think they're important. The nuclear deterrent isn't just some page out of history imo. Yes, as members of NATO we sit under the US nuclear umbrella but a) that makes it into a purely economic decision, which is fine but also b) shows our hand as basically having our foreign policy in lock-step with the US. And for the most part it is, because we're both western liberal democracies with liberal interventionist streaks in the northern hemisphere, two chief members of NATO and the security council and with similar ideological underpinnings. So it's not unusual that our foreign policy is largely the same, but if we simply relied on the US's nuclear arsenal, there would be things that we couldn't even threaten - and I'm not talking about nuclear war - with any expectation of being taken seriously if the US wasn't also playing ball on. Furthermore, we benefit from the umbrella, I've no qualms about funding. Finally, we'd become the only member of the permanent security council without nuclear weapons, weakening our position.