The Christmas song with the initials Z C is "Zither Carol". The song is a Bohemian/Czech Carol. There are at least two versions of the carol.
Nuttin' For Christmas.
The real book that was made the Christmas carol
Walking in the Air.
No, there isn't a Christmas song with the initials TBC. There is one with the initials TBHC (The Boar's Head Carol).
Let it Snow.
Christ is Born in Bethlehem.
O Christmas Tree, (O Tannenbaum), it is German Christmas Carol.
The song that contains the initials OLTOB is Oh Little Town Of Bethlehem.
Every Star shall sing a Carol
Infant Holy Infant Lowly
That would be, "See, Amongst The Winter Snow".