It might not be a bad idea to explain that those links all deal with a completely unrelated song, though.

I was wrong to dismiss this text so readily. It is indeed old; but not particularly so in this form. Quiller-Couch (though he neglected to acknowledge any source) quoted it from Bishop Percy's Reliques of Ancient English Poetry (vol. II), where it was collated, it appears, from the text given in the Prolusions of 1760 (itself a collation of two texts printed in Richard Arnolde's Chronicle, perhaps of c.1521) and variant texts in a further two editions of Arnolde. Quiller-Couch also included the text in his edition of The Oxford Book of English Verse (1900; now deservedly superceded), where he again names no source but describes the piece, without evidence (Percy thought that Arnold(e) wouldn't have printed it unless it was already old, but that is unproven), as "15th century".

The parody mentioned above, The New Nutbrowne Maid, seems to have been first printed by John Scot about 1520. Apparently it was a religious re-write, "upon the passion of Cryste".

So far as I can tell, no tune is known. I confess to seeing no particular interest in this "ballad"; it is scarcely surprising that it did not survive in oral tradition. Percy, however, saw in it virtues which are lost on me:

"The sentimental beauties of this ballad have always recommended it to readers of taste, notwithstanding the rust of antiquity which obscures the style and expression."

Perhaps I am just a reader without taste. Wordsworth, it appears, was impressed by it; but then, he was impressed by quite a few things in his youth (including opium, which explains one or two things) which he later grew out of.