The UK trident missiles are leased from the USA and are manufactured, tested and serviced in the USA (though we do build our own warheads*). TLAM would, if an upgrade was required, similarily have to be shipped back to the factory to have the work done.
*One of the breakdowns in the special relationship was over nuclear weapons. After the war, even though many of the scientists involved had been British, the USA refused access to the technical information. That was the reason the Uk had to set up its own programme and develop its own weapons - a technology it still retains. Exchange of information only, grudgingly, started again after the UK exploded its own H-bomb.