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

Assuming this takes place before the USSR joined the war in June 1941, the war would have been won. Up until that point pretty much all of Europe had been taken over, and so if Britsinbhad not held out until 1941 Nazi Germany would have won the war.

However, to answer your question they would have had an easier time against the USSR and the USA, presuming that Operation Barbarossa still takes place and so does Pearl Harbour.

This is because although Britain was bankrupt(and obviously the land less would have ended) and so had no monetary funds, but its Empire meant that assuming the entire Empire swore aliiegance to Nazi Germany rather than continuing to fight despite Great Britainbitself being conquered, they would have lots of land and some extra troops from around the enpire to that would have benefited all fronts of the war. Also the Germans would have the Supermaribe Spitfires, Lancaster Bombers and Hawker Hurricanes (this is assuming that all British planes are not destroyed during the Blitzkrieg plan) with which to fill the gaps in their Luftwaffe, which was heavily depleted in the Battle of Britain. They would also have the schematics for the Spitfire, allowing them to build more, the Spirfire was a very useful plane.