To begin with, I thought I had it, too, but the John Shiels  Cuckoo's Nest  in the DT is one that I don't know, I'm afraid; though it may be set to one of the well-known tune variants, the file isn't specific enough to be sure; it would have to be stretched a bit to accommodate the lyric!  Frank Harte would know.

While I'm here:

682)  COLLYCW
680)  COLLY MY COW 2
681)  COLLY MY COW 3
Note: For some unknown reason, this transcription by Bruce Olson of a broadside of c. 1675 has been broken up into three separate files.  They should all be together in one file, in this order: COLLYCW (verse preamble); COLLY MY COW 2; COLLY MY COW 3.  Bruce refers to a traditional text and tune in Baring-Gould's Songs of the West; this doesn't comfortably fit the broadside text, and Baring Gould's notes suggest that the tune may not in any case properly belong to the song, as he noted it from "an aged woman at Kingsweare", who was actually singing  The Abbot of Canterbury  to it at the time.  The notes are ambiguous as to whether there was any ground beyond convenience for associating tune and text, and no mention is made of the immediate source of the text, though Baring Gould did note that he had added the final verse himself.  On the whole, I think it best to post the West Country set in a new thread, with references to the earlier version:  Colly My Cow.