The mystery deepens. The Lullay, lullay bit immediately leads me to think of the Coventry Carol which indeed the Voice Squad have recorded (The Voice Squad "Good People All" Shanachie track 6 The Coventry Carol). But the lyrics given on the DT COVENTRY CAROL don't have any 'bone of my bone' type lines. Well, perhaps there's a Wexford version with more lines, or maybe you heard a composite song. The standard Coventry Carol lyrics appear on many websites. Alison provided midi and ABC on a Mudcat thread
There's audial at this carol site
and sheet music at a href=http://www.rememberjosie.org/xmas/s7272a.html
Your best bet is probably the Voice Squad recording of Coventry Carol. Let us know how you get on.
Alice, I enjoyed your link too.