I think that song never existed. I did try searching for it and I only found this thread. P G Wodehouse would sometimes make up things for comic effect. I once search for a long time to find Clarkson's warbler. Which was mentioned in another story. Bertie wonders who is Clarkson, several times so I looked it up and no such warbler exists. A song about an automobile with a honk that goes toot toot, would be a very annoying song so the point, I believe is to show that Bertie is annoying his neighbors and get kicked out of his flat and Jeeves leaves him as well. I almost thought banjolele might not be real, but they are a cross between a ukulele and a banjo.

I also read in some books about P G that he would use the wrong names for flowers in the garden in the wrong seasons. And how there never is a foggy day in London, each day is sunny. He just did not care about details being accurate. But, if that song did exist I would love to get the music and lyrics for it.