The Brand re-write of the Scottish song is quoted in the DT: THE CUCKOO's NEST (2)

The tune link is to the obsolete version of the Fiddlers Companion at Ceolas. In spite of repeated requests from Andrew Kuntz that they allow him to update it, they refused to do so; he therefore moved it to http://www.ibiblio.org/fiddlers/. The old version remains at Ceolas, unfortunately.

The new link would be http://www.ibiblio.org/fiddlers/CS_CZ.htm. Note that the collated information, drawn from numerous sources, is at times contradictory.

So far as the song originally asked about is concerned (the English form, we might call it, though how far the English and Scottish songs are related is worth more investigation), it seems to be a product of the early 19th century, appearing in Fanny Hill's Bang-Up Songster (1835) and on broadsides. See Bodleian Library Broadside Ballads:

Cuckoo's nest

No source of any kind is acknowledged for the DT file CUCKOOS NEST. My guess would be that it derives from the 'Morris On' record, but it isn't a direct transcription; the words have been muddled in places, and 'me' has as usual been substituted for 'my' throughout. My guess would be that the 'Morris On' text was a slight adaptation from a broadside edition, set to the obvious tune, rather than a form received from tradition.

Also at the Bodleian is a broadside edition of THE CUCKOO'S NEST, presumably based upon the earthier song. The DT file is a transcription from a performance by Frank Harte, who said that it was written by John Shiels of Drogheda.

Heaven knows what that cd of 'medieval summer fair songs' was that the original questioner -ten years ago, now- mentioned. I assume that he meant that they were songs performed at modern mock-medieval fairs, presumably in the USA, rather than that anyone might seriously imagine this to be a medieval song.