I like this tune a lot, however, the words ditty and jig apply:

Johny Cock Thy Beaver 

Johnny Cock Up Thy Beaver 

I think this is a tune that would work well in a burlesque of LL, particularly if the point were to ridicule Jacobites, but it might work for a ballad as well. I have heard a Medieval Scandinavian variant with a similiar tempo. Bronson said, "The folk mind- if the term be allowed- has never distinguished ballad tunes from any other good singing tunes."

But I still wonder why there is no listed variant Johnny O'Cockelsmuir. The only place that name ever appears in a search is in Percy Society publications. Since it is common in Scotland, this surprises me. The fact that it does exist does not change my opinion that the tune was deliberately assigned to LL by the Northumbrian balladeers, Dixon et al.