The tune on HTTW's website is the one I know as "Haste to the Wedding" and I have seen it in several publications under that name.

The fiddlers companion has this to say about it:

The tune 'Come, Haste to the Wedding,' of Gaelic origin, was introduced in the pantomime, The Elopement, staged in London in 1767. This version is known as the Manx tune and was printed by the Percy Society in 1846. It is the basis of the Manx ballad, 'The Capture of Carrickfergusby,' written by Thurot in 1760" (Linscott, 1939). Samuel Bayard (1944, 1981) comments on the popularity of the air over the past two centuries as well as the tenacity of the main title to stick with the tune. When Chappell printed his well-known set in National English Airs (1840, I, No. 163; notes, II, 129; reprinted in JEFDSS, III, 210.), he traced the tune to the year 1767, when it was used in pantomime, to a song beginning 'Come, haste to the wedding, ye friends and ye neighbors!' This version of the air, continues Bayard, is still the earliest known, and it may be that the popularity of the song occasioned the fixed quality of the title. In his 1944 work Bayard posses the question as to whether the words were included in the original pantomime as a result of its associations, or whether the later uses of the tune secured it...

It also gives abc's of three variants.

X:1
T:Haste to the Wedding
M:6/8
L:1/8
R:Jig
B:Sharp – Morris Dance Tunes
Z:AK/Fiddler's Companion
K:D
DFA A2f|ede fdB|ABA AGF|GFG E2F|DFA A2f|ede fdB|AFA faf|ded d3:|
|:afa agf|gfg bag|fga agf|gfg efg|a3 f3|ede fdB|AFA faf|ded d3:|

X:2
T:Haste to the Wedding
M:6/8
L:1/8
K:D
DFA A2f|ede fdB|ABA AGF|GFG E2F|DFA A2f|ede fdB|AFA faf|ded d3:|
|:afa agf|gfg bag|fga agf|gfg efg|a3 f3|ede fdB|AFA faf|ded d3:|

X:3
T:Haste to the Wedding
M:6/8
L:1/8
R:Jig
B:Stewart-Robertson – The Athole Collection (1884)
Z:AK/Fiddler's Companion
K:D
B|AFA Aaf|ede fdB|AFA AdF|EFE E2A|AFA Aaf|ede fdB|AFA faf|ddd d2:|
|:f/g/|faf faf|bgb bgb|afa agf|efe e2f/g/|a3 f3|ede fdB|AFA faf|ded d2:|