Danny Couper sang a Scottish version of this [ which he got from Jeannie Robertson ] at a ceilidh in Aberdeen last night. The tune he used - essentially the same as Carthy, but a slightly different rhythm - he called "The Rose Tree", which I think is the Scottish name for it.
The tune they used for "Four Nights Drunk" is certainly sometimes called "The Primrose Lass" in Irish traditional music circles.