Some notes on the list of settings I posted:

The Supply Belcher setting was titled "Carol".

"Flensburg" was written by Louis (Ludvig) Spohr (1784-1859) and harmonized by Joseph Barnby, 1867. You may hear it here: "My God I Love Thee"

"Lydia" was written by Thomas Phillips (1735-1807).

In the Hymnary.org list, "Shepherds" is Sir Arthur Sullivan's arrangement "Bethlehem" of the traditional "Gabriel". "Noel" is his arrangement originally for "It Came Upon the Midnight Clear". The tune is also known as "Eardisley" or "Gerard", probably a variant of "Dives and Lazarus".

"Hampton", written by McNeil Robinson II, b.1943, appears in The Hymnal 1982, #95.

"Hitchen Carol" is an English traditional melody, but I found no more about it.

"St. Magnus" is indeed Clarke's "Angel's Carol", first published in John Playford's Divine Companion, 1707; it was originally used for Psalm 117, and was first combined with the "While Shepherds" text in the 1868 Appendix to Hymns Ancient and Modern.

"St. Martin's" was written by William Tans'ur, 1740, and is used for at least six other hymns.

"St. Ursula", by Frederick Westlake, Hymns and Sac­red Songs for the Year, Part I, 1863.

"Southwell", by Herbert Stephen Irons, 1861. May be the same as the Irons arrangment listed previously.

"Nottingham" = "St. Magnus"/"Angel's Carol" (Jeremiah Clarke)

"Christmas Bells", from "The old year's long campaign over", by Samuel John Stone (1839-1900).

"Northrup", by Abraham Northrop (1863-1939), by 1906.

"St. George" may be the one written by Henry J. Gauntlett, 1848.

"Lyngham [Desert]", by Thom­as Jar­man, cir­ca 1803.


Additions:

"Sweet Chiming Bells", the Tate text with an interpolated and unrelated chorus.

William Billings wrote two settings:
"Bethlehem", The Singing Master's Assistant, 1778.
"Charleston", 1770, titled "A Hymn for Christmas".
He also wrote another setting "Emanuel" to a text of his own, a fanciful revamping of Tate's text.

"New Bethlehem", by Edward French, The Stoughton Musical Society's Centennial Collection of Sacred Music, 1878.