I listen this version on Johnny Cunningham first album "Thoughts from Another World", but I don’t know if the B part is traditional or original beacuse I never ever listen that melody set to the classic Christ Child Lullaby.
Stumbled upon this version which I really like. They sing it in C mixolydian and I have transcribed it in the same key. It lies easily on the fiddle. Here it is in G mix for flute and Uilleann pipes.