Think about u. s participation in the war in europe. how might the axis have benefited by gaining control of great britain by 1942

In 1942, Great Britain was at the head of the British Empire, the largest Empire in  known history.

At it's peak, the empire consisted of nearly 25% of the world's population, a large landmass, unmatched resources and hundreds of millions of subjects.

If the axis were able to control Great Britain, they would quite possibly have complete hegemony over the world.