the Shorter New Oxford Book of Carols (1993) notes the following:
"The origins of this tune have yet to be completely uncovered (the attribution to Handel probably results solely from the resemblance of the opening to 'Lift up your heads' in Messiah). It first appears as an English psalm tune in the 1830s, and was set to Watt's text, a paraphrase of Psalm 98, by the American Presbyterian Lowell Mason in 1836."
I think three of the first four notes are the same, but that's pretty much it.