Barley Mow is mentioned in Olson's website, from "Thompson's Compleat Collection of Country Dances, vol. 4, ca. 1780.
Some contradancers use the tune Linden Hall.

Nothing to do with the origin, but a nice little verse is inserted into the routine by Billy Leatherbreeks called "Bobobella; or, The Blacky Moor Keelman" (pub. 1870s in "Tyneside Songs, by E. Corvan and G. Ridley et al.), tune for this segment "Dixey's Land."

"..... On, on, wi' yor sprees, let the gam gan on now;
An' end a' yor glories i' the aud Barley Mow.

"Aw wish aw was i' the Barley Mow,
Wiv a quairt o' yell before me now.
Huzza! huzza! huzza! huzza!
For iv a' the places aw ever met,
Thor's nyen can beat aud Sangyet yet.
Huzza! etc.

Chorus
"So drink away mee hearties
Huzza! etc.
Niver say die, for that's all mee aye,
So drink away mee hearties,
Huzza! etc."