It seems to be William Chappell in his Popular Music of the Olden Time who claims it is included in The Shepherds' Oracles. James Hogg in Jacobite Relics apparently claims it is an anti-Whig song.

Chappell does so state (p. 425), and his nine verse version has the same verses as those listed here. He states as other sources MSS. Ashmole 36 and 37 (the Ashmole collection is in the Bodleian Library, parts of which have been digitized, so you might be able to find scans); Loyal Songs written against the Rump Parliament, Ellis's Specimens, and Stafford Smith's Musica Antique, with a different tune.

There is also a Bodleian broadside, Bodleian, Vet. A3 c.29(6). This one hints that the tune was already known by the name "Hey Then Up We Go" before the words were written.

It's Roud's V19592. He has a couple of other listings under different titles.

Bruce Olson listed dozens of other songs to that tune, but they seem almost exclusively to have been topical and forgotten.

There were no Whigs in the 1640s, and Hogg actually had the date right; he wrote ""I am informed ... is one of Charles I.'s time, and that it was originally an English song, though popular in this country."