"For example, iran nukes uk and says if the us intevenes, NYC is next but promises if the US stays out, it is safe. This puts the US in an uncomfortable position. You having nukes is the best deterent against being nuked."
But, as pointed out, Trident is not an independent detterent (as the French nukes are, for instance). Your hypothetical example works just as well with Iran threatening to nuke NYC, but promises if the US does not allow the launch of Trident it is safe. Same difference.
Why is Trident not independent?
There is a difference between the US saying "go ahead" and the US actually doing it.
You have a better deterent with a trident than without.