There's more on The Bitter Withy / The Holy Well in earlier discussions, chiefly History of 'Mary Mild'.

It wouldn't be at all surprising if On Christmas Day derived from the homilectic chapbooks that circulated widely in the earlier part of the 19th century; as, it appears, did The Holy Well, whatever its antecedents may have been.

I should mention that Fred Hamer's recording of May Bradley can be heard on A Century of Song (EFDSSCD02, 1998), a CD of archive recordings isssued by the English Folk Dance and Song Society to celebrate the centenary of the Folk Song Society:

http://www.efdss.org/century.htm