A version will be in the next (8th) volume of 'The Grieg-Duncan Folk Song Collection', due out later this year. I have what's there, but can't divulge it prior to publication. (I'd heard that it was in the Greig-Duncan MSS, but when I didn't find it, I enquired about it from the editor about a month ago.) Somewhere I've got a xerox of a version one published by Ruth Tounge(?) in England some time ago. I think song and tune are really English, and the well known tune is at least one Scots collection of the 18th century. I don't think I've seen the tune in any old Irish collection of popular tunes.
Look in the Opie's 'Oxford Dictionary of Nursery Rhymes' for a fragment.