This is presumably the same as the naval component - the missiles and warheads are American, and I'd be amazed if the UK can use them if the US vetoes.
Trident carries UK-manufactured warheads and is allegedly independent, even though it does rely on US ballistic missiles to carry them. Wikipedia says -
"The British government insists the warhead is indigenously designed, but analysts including Hans M. Kristensen with the Federation of American Scientists believe that it is largely based on the US W76 design"
There appears to be a new design in the works at the moment called "Astraea"
