The idea that the Western Allies waited until the USSR had effectively defeated Germany before walking over Western Europe was a actually the party line and pop history taught in the USSR and the nations on the other side of the Iron Curtain during the Cold War. I wouldn’t say that the contributions of the USSR were down played in serious history during the Cold War, but they certainly were minimized in pop high school history if what I was taught in high school is any measure. It consisted largely of mentioning the German invasion, Stalingrad, and the end of the war, while spending a great deal of time on what the Western Allies were doing. Normandy alone received more attention than the entire Eastern Front.
Britain was already involved before the attempted invasion. Poland was the line in the sand which Britain and France determined that Germany had to be stopped. I can’t imagine any case where Britain would have quit its war against Germany, especially once France was invaded. That would have been suicidal. Germany was clearly the threat to Britain, and there is no basis to the speculation that Britain would support the fascists against the communists.
However this ignores the factor that it was the Japanese generals spoiling for the fight, which let to the threat to US interests. While the US would not have gone to war as soon as they did without Pearl Harbor, what reason would they have for giving up the embargo on oil and scrap metal, the reason behind the Japanese military’s attack on the States and Britain.