My favorite site for information about music, musicians, what-song-appears-on-what-album, is this:By searching for "Willy o' Winsbury" I found that it is performed by Sweeney's Men on the compilation album Irish Folk Favourites, Vol. 2 [St. Clair]. Unfortunately, that's the only listing it has for this song (or tune?).
By following up links, you can then see lots of information about Sweeney's Men -- for example, band members' names, albums, and similar artists.
If you click on an album title, you SOMETIMES get a list of tracks, along with composers' names. This is more likely to happen if the album is currently in print.