I don't think the UK can launch without the permission of the US.
Having a read there, the Trident missiles themselves will be able to be used right into the 2040s. The submarines are what needs replaced imminently, but despite the US has extending the life of their own boats by 10-15 years, the UK gov seem hell bent on spending a fortune on new subs.
Personally if the UK must have a nuclear arsenal I'd rather it was a more strategic system like cruise missiles than the Cold War relic that is Trident....