The UK has always designed and maintained it's own.
Trident may share it's technology and "blunt end" with it's US brethren, but the bit at the pointy end is designed and built at that place in Berkshire.
Not since the early 1960s have the UK designed and produced a nuclear weapon physics package on their own (the fisson trigger for WE-177 was the last). Yellow Sun Mk 2 was a British copy of the US B-28 design, The warheads for the SSBN's were based on designs used on the US warhead designs, though the WE-177 primary (atom bomb trigger) design was re-engineered for the Polaris warheads as the UK thought the High Explosive used in the US warhead for their missiles was unsafe. Its a good bet that the majority of the bits in the UK Trident re-entry vehicles were made in the USA.