I've no idea whether it's got anything to do with Lord Randal. I was made aware, from my reading, of the idea that 'Billy Boy' and 'Lord Randal' are sort-of counterparts to each other, humorous and tragic, and both have relatively old antecedents.
The Note in Bewick merely hints at this muddy area, it doesn't have answers. But there is a good tune included, and it fits the Randal words rather better than the Billy words. It's all a big muddy soup, and what does it all mean, I ask you, eh? -- To get on or off this list see list information at http://www.cs.dartmouth.edu/~wbc/lute-admin/index.html