It's not as straight forward as asking permission, but you should read about Trident as a missle platform.
To say the UK has a genuinely independent nuclear deterent is not as clear cut as you'd think. But that's not the same as being absolutely beholden to the US.
That all said, we really do need to swap to the french missle system, but that will take time to retro fit the subs.
Thanks, so it's cleaner rather than clean. Makes sense.