Mulberry Bush I'm afraid.
No indication in Karl Dallas's book of any previous publication. In his note in the book he says "A song from the singing of Mary Reynolds of Co Leitrim in Ireland...Some have called the melody trite and 'Irish pub tenor', but these words and simple tune are strangely moving, especially as Mary sings them in her high sweet voice." Which may sound a bit patronising, but seems to suggest that he might well have collected it himself. Or maybe it's on a Topic record or somethinhg like that.
The fella who might know about this, since he knows about everything to do with Irish songs, could be Frank Harte, who sometimes frequents these Mudcat parts. I've sent him a PM drawing his attention to this thread which he might see.