In the earlier thread Kitty indicated, the late Bruce Olson quoted a London broadside of the late 17th century, 'The Unkind Parents', which is an early ancestor of this group of songs. That isn't too bad as the beginning of an answer, surely? More detail can be seen in Steve Gardham's article at the Musical Traditions website:

The Turtle Dove.

The song reappeared in the 18th century, rather shortened and closer to the kinds of forms it was found in later, as the broadside ballad 'The true lover's farewel', which is directly ancestral to traditional examples found in such places as Dorset and the Appalachians. There was, incidentally, a tradition in the Traill family of Orkney that their ancestor, Colonel Thomas Traill, wrote the original song in the earlier 17th century; but there are many such family traditions, and usually no evidence (as is the case here) to support them.

Originally an English song, in view of the available evidence; though Scotland may not be quite out of the running. Irish? No, though of course forms of it have been found in tradition there, as also in pretty well every country where English is spoken. I'm talking about the text here; there are many tunes associated with this song family, and they would be the subject for a different (and much more technical) discussion, I think.