If our nuclear deterrent were ever used in anger then it would be "American" missiles that would be fired.
While the technicalities of the missiles performance might well be an American responsibility, the ultimmate success or failure of the mission is very much a UK concern.
If the telemetry in a test missile fails, it raises the inevitable questions about the reliability of the components of a war shot missile (yes, I know war shot missiles aren't fitted with test telemetry).