Happy to learn otherwise but I think the reason why people say the UK nuclear deterrent is not independent is because they assume it uses US satellites for navigation and targeting, although it actually uses starlight, night and day.
Obviously long-term the UK would rely on US technical support, but not initially.
Although the UK is kind of dependent in one way because if the US did launch then their enemy would likely hit the UK anyway just in case. But in the ludicrously unlikely scenario where the UK wanted to bomb someone the US didn't, it could, if that makes sense.