In fact, I could make the argument that England has long been invaded by the US since Suez crisis, and is in fact, the 51st state of America. Most of England's foreign policy in recent years appears to be dictated by the white house anyway.
You could, but you'd be wrong. US "dictating" UK foreign policy was basically just a matter of Blair sucking up to Bush; you'll note that Britain kept out of Vietnam, and in the run-up to the Gulf War Thatcher was pushing Bush Snr. to undertake military operations, rather than vice versa. And even ignoring this, your argument rests upon a conceptual confusion. Being dependent on another state is a common outcome of being invaded, but this doesn't at all mean that being dependent on another state is
the same as being invaded.
UK forces are part of NATO, the US is the head of NATO. surely the US would have some idea where the UK's nuclear submarines are? it probably already keeps a tab on UK's nuclear subs, and this prior knowledge may not be for the general public to know.
Unless the US has somehow managed to sneak tracker devices onto all Britain's nuclear subs, no.