James Barke cited "Pop goes the weasel" as the tune and gave a very slightly divfferent version of Kiwi's song above in his preface to the Barke/Smith/ Ferguson edition of 'The Merry Muses of Caledonian'. I don't have the original 1959 British edition, but the 1964 American one, where the song is on p. 28. He said the song was from school playgrounds.