I see no reason why the same tune wouldn't have been used both sides of the border. The ballad was obviously very popular in London, and by 1600 England and Scotland, though it was another century to the Union, were inextricably linked by intermarriage of the nobility if nothing else. Not long after 1700 Ramsay was publishing books chock full of London theatre and pleasure garden songs. 'Scotch' songs were very popular in London in the 17th century, albeit mostly imitations.

By Ritson's time just about all traditional ballads had fallen out of usage in England, or were certainly off the radar. Even Percy's Reliques which set Europe afire and had its effect on Scotland, moved very little in England, except as reading matter for the well-off.