I missed that one first time round, too; here is what Iona and Peter Opie have to say about it in The Oxford Book Of Nursery Rhymes (1951):

This rhyme, to which there are numerous appendages, is well known in the present day, often accompanying a hand warming exercise in the manner of "Pease Porridge Hot".  Its literary history, however, is strangely short.  It is first found in Come Hither (Walter de la Mare, 1922).
Subsequently it was discussed in Notes & Queries (1931-4) and the absence of 19th century recordings was remarked upon, though it had been known by correspondents sixty and seventy years previously.  One writer declared that his grandfather, born in 1818, had learnt to play it on the violin, when a boy.  "It was then, and is still, a favourite at village weddings or merry-makings whenever a lively jig of the polka type is needed."  The tune is identical with the first part of The King Pippen Polka (c. 1870), and has also been recognised in a gypsy dance in Smetana's The Bartered Bride.  Versions of the rhyme repeat fragments of singing games, and it is possible that it was once part of some dramatic "joie de vivre."

Apparantly there was an advertising slogan of 1946 based on the song:

My mother said that I never should
Miss my Guiness -as if I would!

Malcolm