Jesus, not this again. How many times does it need to be said? The UK does not require the permission of the US to use our deterrent, the UK does not need launch codes from the US and the US doesn't have a cut off switch to prevent the UK using it's deterrent.
In the unlikely event that the UK looses off all it's missiles, yes, the US could make life difficult then by refusing access to Kings Bay where the common pool of Trident D5 missiles are stored. This is however a bit of a mute point as the UK has ownership of 58 missiles and circa 200 warheads distributed in four boats, 8 missiles per boat, 6 warheads per missile for a total of 196 warheads deployable at any one time, so there'd be little point picking up the missiles in Kings bay as we'd have little to put on them, at least in the short term.
However if it came to that there'd be bigger issues, like the continued existence of the United Kingdom, to worry about.