I have an Andrews Sisters recording of this song from 1958 that credits A. Gershunoff and E. Pellegrini as writers--but Alex Gershundoff was studio musician, a sax player, I think, for the NBC studio orchestra and Capitol Records, at that time- However,I have read that it was an old music hall tune, written by either Weston and Lee or by the same people who wrote "Istanbul (not Constantinople)--

The melody, curiously, sounds like it is a Turkish classical melody--