"The Boar's Head Carol" in its separate forms has no line of descent from pagan times- that is just more fakelore.

The various forms and the university legends (16th c.) and the earlier carol by Smert(?) are discussed by Masato Sakurai and Liz the Squeak in thread 27543: Boar's Head"

Abstracting from that thread-
The earliest record is titled "The Borys Hede That We Bryng Here," attributed to Richard Smert, rector at Plymtree, 1435-1477, from Die Nativitatis, a MS collection from the 15th-16th c. and printed in Ritson's Ancient Songs; the carol later published by Wm. Sandys, 1833.

See article by William Studwell, The Christmas Carol Reader, reproduced in: Boar's Head Carol
(A link at the end of discussion of this carol will lead to "The Borys Hede--."

The website, "The Hymns and Carols of Christmas," is the largest online, and has discussions of many carols and hymns, as well as history.