In the beginning credits it states the name of the song, "Were You Smiling at Me" and directly underneath is Melvyn Martin's name. So I guess we can assume that he wrote it? I've tried to find it in various places too without any luck - but at least it is always available to us within the film.
He's not in the ASCAP Biographical Dictionary - and you only need to have one song performed in public and/or recorded to become a member - so possibly it was a one-off.