My sincere thanks to John Moulden who posted the below response to my question - I have been waiting years to find someone who knew this song and was rewarded by his kindness - thanks John! Mary-Ellen

"At the time this query was posted, it rang no bells but now, serendipitously, I have found a version.

It's in James N Healy: The Mercier Book of Old Irish Street Ballads (vol 1) ; Cork 1967.

On page 59 is "A new song called the lamentations of Jane M'Cullen For the murder of her child" 8 stanzas, 64 lines in total. It is the song you describe. Interestingly McWilliams and McCullen are not far apart - here in co Antrim McWilliams is sometimes said "McCullion."

The only place name is "Clifton" presumably, but not proveably without research, Clifden in west co Galway.

Healy gives no sources for the text but, in common with most of his texts, it is most likely from a ballad sheet in the National Library of Ireland. His texts are sometimes tidied silently.

If you can't get hold of the book through a library, let me know if I can be of further help."