There's another in DT. DT's WARNWILI is 'Scots Musical Museum', II, #188, where it appears with the tune. This Robet Burns recasting of a song in 'The Charmer,' 1752, into on on the rebellion of 1715. I don't have a complete copy of The Charmer text, but the 1st verse is the same as that in James Hogg's 'Jacobite Relics', II #5, 1821, and has the same number of verses, so is probably the same. In his notes Hogg says that the song was probably woven around the chorus of some older song, and that there was no 'Willie' of note in the whole Jacobite army (but what rebellion is he talking about?). The tune is found in Walsh's Caledonian Country Dances, bk. 3, c 1740.

There's an entirely different song in a chapbook, 'Three Excellent New Songs', c 1746, and it's not from the Jacobite, but from the Loyalist side:

"The Loyal Subject's Joy for Duke William's Victory over the Rebels". This starts:

Now tune your Pipe, and dance your Fill,
Wi Mirth and meikle Glee, Laddie
For Cumberland is now come down
Frae Rome to set us free, Laddie
[Cho:]Up and war them a' Willie,
Up and war them a' Willie.
Fair fa' the Lad that fears nae Skaith
'Mang Dales and Hills of Snaw, Laddie.

In NLS MS Adv. 19.3.44 is a song of 11 verses which commences:[Chorus first]

Up and rin your' wa' Willie
Up and rin your wa's
Cullodens Lawrells you have lost
That gain'd you such applause
A check of conscience for your crimes
That stings you to the Saul willie
Has broke your measures this Campaign
As much as Lewendhal's willie
Up and rin, &c.