Is "I'll Walk Alone" different from "You'll Never Walk Alone"? (YNWA is from the right time.)
Is the George Formby song about the Maginot Line "We're Gonna Hang Out the Washing on the Siegfried Line"?
"Roll out the Barrel", aka "The Beer Barrel Polka", was popular in Britain in 1939-40 and also in the U.S. in 1940, before the U.S. entered the war. According to The Timetables of History,, where I'm getting all this information, it's of Czech origin.
Another (and very moving) one from 1940 is "The Last Time I Saw Paris".
"Rudolph the Red-Nosed Reindeer" is a few years later.
Oklahoma is from 1942. You probably don't need me to suggest "People will Say we're in Love" as a duet, Vixen.