I think it's the other way round, Liz. The Ernest Jones poem is very close to the text posted above, so I would think yours is the rewrite. The original is printed in Pinto & Rodway: The Common Muse. The poem (c.1848) has the note: Music by John Lowry. This song can also be sung to the air of "The Monks of Old" but I don't know anything about either tune.

The main textual difference is that Martin Carthy changed the phrase "we're so very, very low" or its variants to "we are so low," presumably to fit his music better.