Given that the nukes that the UK has is Trident, which is a US system that the UK cannot use without US cooperation [0], it seems entirely appropriate that the USA gets to decide if the UK has nukes.
[0] Yes, the UK can fire them without US approval, but the actual hardware is maintained and supported by the USA, and they have to be shipped to the USA regularly for maintenance. If the USA decided that the UK should not have nukes, there's not a lot the UK could do about it, the Trident system would have to be scrapped entirely and replaced with some completely different system. Which the UK doesn't really have the capability to do and it would cost a fortune to acquire that capability.