As MGM Lion found, there's a 1949 recording by Dan Dailey and the Andrew Sisters. I always thought this was a music hall from the 1920s, but maybe the Andrews/Dailey recording was the first performance (although Jim Dixon found a 1941 publication date).

Here's a pretty good recent version by Shanneyganock: https://www.youtube.com/watch?v=FSk0SxaXgbw