The UK did take some advantage of American components, the engine programme for Blue Streak started with licensed American designs that were progressively replaced with indigenous designs. Black Arrow was a purely British development. British companies had some rocket experience because in the 1950s there were several plans (Saunders-Roe SR.53 and SR.177, Avro 720) to use rocket engines to boost military aircraft. This development was superseded by usable afterburners, but not before several rocket engines were developed.
The giant radio telescope at Jodrell Bank played a role in the early days of the space race, tracking Sputnik and others via radar, and assisting with (IIRC) some of the early US Moon probes. By the time Apollo started, NASA had enough large antennas of its own and Jodrell was no longer used officially (they still pointed the antenna at the Moon for the Apollo 11 landing and listened in on its communications, IIRC).
France did not use American technology, Diamant was a purely French development.
Both nations' rocket programmes started as military programmes, with the desire to have a nuclear deterrent independent of the Americans. That meant occasional reinventions of the wheel were inevitable.
There was at least one European contribution to Apollo in the shape of scientific experiments. I don't believe any Apollo spacecraft hardware was built in Europe. NASA did buy some loose items of equipment in Europe though: NASA used Hasselblad cameras (standard cameras with some modifications) and Zeiss lenses (some standard, some specially designed for NASA) in the Apollo program.