We've been watching the re-runs of Jeeves and Wooster on Sky, and the above song appears in at least two episodes. For some reason, we liked it, and wondered it if was a real composition, or something made up for the show.
To my delight, I found out that it was indeed written and composed by Leslie Sarony in the 1920s, and there's a copy of the dots on the National Library of Australia website.

Guess what I'll be learning next...