A lot better information is in the Opie's 'The Oxford Dictionary of Nursery Rhymes'. The Opie's don't buy the romantic nonsense unless there's evidence for it, and most of the stories about the personages in nursery rhymes come from overactive imaginations. One sees the same thing in folksongs. Spinster daughters of ministers seems to be best at locating some hidden religious mysticism in old songs that don't have a well connected narative.

I have an early transcript of "Willie Wastle". That in DT is by Robert Burns (from Scots Musical Museum, #376), and its tune (not in DT) is "Sic a wife as Willie had"