We've had that discussion, during the 80's and 90's we reduced the total of world nuclear weapons from 26,000 to around 18,000. It's been at that number ever since.
Everyone gets the sentiment about Trident and nuclear war skewed; no, nobody wants to suggest using Trident. YES, if possible we would want all nuclear weapons to be destroyed, however nobody is willing to get rid of ALL of them whilst their enemies in the world still possess the knowledge to make one and just one can do untold damage.
Knowing that if you use one on your "enemies" would result in the complete annihilation of your nation and people is a powerful defence tactic. It's why one hasn't been used although often threatened. Nukes come in many forms, not just ICBM or IRBM's. To use a simple analogy, you and your neighbour are having a dispute; you both know you both have guns. After years of fighting and threatening to shoot up each others houses you have dialogue to each get rid of your guns. You both agree, in front of a neutral party, to get rid of your guns one by one until they are all gone, yet after week 4 your neighbour hasn't gotten rid of some of his weapons, leaving with him having more than you.
Do you continue to get rid of yours as per the agreement in the hope your neighbour will get rid of theirs, or do you wait until they match your total of weapons?
What happens when you both get down to one gun each; who will be the first to get rid of theirs? How do you know that the moment you get rid of your gun, he won't pull a Dead Man's Spin on you? You don't, and the world isn't at that stage of total trust either, so if your kids ask you if you would use your guns to protect them from the neighbour next door, what would you say to them to give them confidence you can protect them? All he'd have to say is "i'd never use them offensively, but if our nation and citizens were being threatened, i'd have no hesitation in doing what is necessary to prevent such an eventuality." But he's not, he just keeps saying what we already know; nuclear weapons are bad, nuclear war is bad and we want to get rid of them. We know that, but we're not asking what he'd do about Trident in the future, we're asking if he'd be prepared to make the ultimate decision to utilise out ultimate deterrant in the face of the ultimate threat and so far his response has been ... we'd need to talk. Not exactly filling the public with confidence.