The song "How Could You Do That to Me"," performed by Rosemary Lane and Johnny Downs, supposedly takes place in the 1930's and represents a vaudeville act they performed in the 1920's, but the lyrics mention World War II-era gas rationing.
A nightclub comedian does an impression of James Cagney in Yankee Doodle Dandy, which was released in 1942, though the scene in question takes place in 1935.