Joe, hymn tune names, like song titles, aren't always unique, particularly between different regions or congregations. If you'll notice, the Pentonville tune you linked (for the hymn "Like Sheep We Went Astray") was written by Frances Linley, while the Pentonville tune which the Boden entry linked to was written by William Marsh, late 18th c.

As far as tune names for various settings of this text, here's a list to get you started:

"Winchester Old", after Christopher Tye; Este's Psalt­er, 1592; the most common tune nowadays [NOBC 46.I]
"Chestnut" major-mode variant of "God Rest Ye Merry Gentlemen" [NOBC 46.III] (1)
"While Shepherds, (Old)", aka. "Ould Zaddok"; coll. Ralph Dunstan, 1929. Supposedly very old.
"Crüger", Johann Crüger, 1657.
"Christ­mas", George F. Han­del, 1728. [NOBC 46.II]
Psalm 8 setting by Michael Beesley, 1748; used by 1760s. [NOBC 46.IV]
"Crowley", Psalm I in Evison's Psalter, 1751; coll. Ralph Dunstan.
"Lob Gott, ihr Christen all gleich", by Nic. Herman, harm. by J.S. Bach
"Sherburne", Daniel Read, 1783; in The Sacred Harp, 1844.
setting by Supply Belcher, 1794.
"Martyrdom", Hugh Wilson, 1800.
"New Tregoney", ca. 1800-1810; coll. Ralph Dunstan.
"Cranbrook", Thomas Clark, 1805; used for "On Ilkley Moor Baht 'at" [NOBC 46.V]
"(Old) Foster", John Foster, ca. 1820. [NOBC 46.VII]
minor setting from Davies Gilbert, 1822.
setting ca. 1830, pub. Ralph Dunstan, 1925. [NOBC 46.VI]
setting by Alexis Theodore Lyoff, 1833
"Zerah", Lowell Mason, 1837.
"Bethlehem" by G. W. Fink, 1842.
"Carol", Richard Storrs Willis, 1850 = "It Came Upon the Midnight Clear"
setting from John Clark Hollister, The Sunday-School Service and Tune Book, 1863/5, #9
"Glory Shone Around", shape note setting
"Bethlehem", trad. arr. Sir Arthur Sullivan, 1874; tune aka. "Gabriel"
"Noel", trad. adapted by Sir Arthur Sullivan, 1874 for "It Came Upon..."
"Shackelford", Fred­er­ick H. Chees­wright, 1889
setting or arrangement by H.S. Irons, 1894.
setting by B.S. Barclay, in the American Memory online collection.
"Vision of the Shepherds", A.P. Howard, by 1916.
"Angel['s] Carol" by J. Clarke, by 1918. [?= "St. Magnus"]
"Lydia", from W.A. Pickard-Cambridge, _A Collection of Dorset Carols_, 1926.
"While Shepherds, (New)", coll. Ralph Dunstan, pub. 1929.
"The Song of the Angels", William Knapp.
"Shining Star", Terry Wootten, 1988; The Sacred Harp, 1991.

Hymnary.org also lists these:
FLENSBURG [by 1874]
GABRIEL (trad.)
HAMPTON (Robinson)
HITCHEN CAROL
SHEPHERDS (Sullivan) [?= "Bethlehem", aka. "Gabriel"]
ST. MAGNUS (Clarke) [?= "Angel Carol"]
ST. MARTIN'S (Old)

One of the references given in the Wikipedia entry also lists these:
St. Ursula
Southwell, by 1871
Anglia, by 1874
Nottingham, by 1874
Christmas Bells, by 1917
Northrup
St. George
Lyngham
Hampton

Of course, some of these entries and names may refer to the same tune.

NOBC = The New Oxford Book of Carols (Keyte and Parrott). The authors mention that the 1708 edition of the Supplement gave 75 tunes for use with 12 texts and 26 alternative psalm translations. "While shepherds watched" was there set to "St. James"

"but with the rubric 'or any other tune of 8 and 6 syllables [i.e. in common measure]' ('Winchester' is given with Psalm 84, 'O God of hosts, the mighty Lord'.) The now familiar conjunction of tune and text is found in Hymns Ancient and Modern (1861)."

Thus they opine that people may have sung the text (in mix and match fashion) to the "Old Winchester" tune as early as 1708. It is unclear if the "St. James" setting was given previously, particularly with the text's first appearance in the 1700 edition of the Supplement. In any case, "Old Winchester" doesn't appear explicitly paired with "While shepherds watched" until 1861, and the authors further state that "Chestnut" was the most popular setting prior to the 20th century.

To give slightly more info on the tune "Christmas", it was adapted from the aria "Non vi piaque ingiusti dei" in Händel's opera Siroe, Rè di Persia. Keyte and Parrott suspect that Lowell Mason prepared the arrangement.