Control was handed to Washington when the decision was made to use a missile delivery system designed, manufactured and overhauled in the US. Even submarine-launched test firings are conducted in American waters near Cape Canaveral under US Navy supervision.
It is inconceivable that Number Ten would fire Trident in anger without prior approval from the White House.
Persisting with Trident and its proposed replacement in order to retain a permanent Security Council seat is to reject British pragmatism in favour of la gloire. At least the French went to the trouble of developing their own submarine-launched missile delivery system. They own it, hence control it.