If you know it, then why do you think that Britain could have carried out Gymnast without US involvement?
They certainly were preparing it. If the US is not in the game, then neither is Japan, which gives UK quite a few free divisions and ships. Once more, RN controls the Med. They can send more reinforcements and faster, as well as supply their people in NA and, at the same time, deny this to the Axis.
The invasion of Europe is something else entirely, but anywhere overseas, the British have upper hand...