I have just spoken to my mum (84, and hating it!) and she says she knew it as a long-rope skipping rhyme around 1925. (It's obviously older than that. To me it feels Edwardian). Now at least I know who taught it to me! She says she never knew any more of it, and that to her it seems complete. She also says it should be: "with a telegram to say that he couldn't come that day/ and they'd find him in the river with his toes turned up."
Probably the reason I don't know the tune is because my mum must have known it as a sort of chant, and anyway is a far worse singer than me. She avoids any form of singing as she cannot sing a note in tune at all. (I occasionally manage one or two).
Andrea