Child published one tune for "Willie O'Winsbury". It can be found in this version of volume V at Gutenberg.org:
https://www.gutenberg.org/cache/epub/71104/pg71104-images.html#INDEX_OF_BALLAD_T

(If the link fails you can cut and paste BUT you must change the last T to TITLES.)

It is the third tune on page 418. The link lands you at the start of the ballad titles index; you have to scroll up from there, watching for page numbers to the right. If you happen to go to the top of this (long) page, do not be put off by the fact that the tune section is not included in the table of contents - it really is there, just above the title index. And there is a LISTEN link so you can hear the tune. Apparently it is the tune for Child's J version. (I use this tune even though I don't sing the J version words.)

It is credited to "Miss M. Macmath", sister to William Macmath, one of the Scottish collectors who provided many ballads to Child.

Frequently the use of the "Fause Foodrage" tune in modern times is credited to Andy Irvine.