Parallel Anthology Track 10
Track 10: Burnett and Rutherford - Willie Moore Columbia 15314-D; Recorded in Atlanta, Georgia, 3.11.1927
No image of record label available. www.78discography.com does not assign authorship, nor does either Anthology booklet.
The reissue anthology booklet, however, states that "Burnett remembered learning ‘Willie Moore’ from a printed ballad".
Lyle Lofgren states that, "The words were from a printed broadside (called a 'ballot' in the mountains), and the music is the fine part (the high part) of an old ballad tune that was floating around Kentucky." 
The Where Dead Voices Gather blog states that "'Willie Moore' is the first track on the Anthology to originate in the United States. Burnett later recalled having learned the song from a printed broadside in his native Kentucky. According to Smith's liner notes, a Mr. Paul Wilson of Farmington, Aarkansas reported meeting a Rev. William Moore in Dallas, Texas, who claimed to be the inspiration for the song (reported in Vance Randolph's Ozark Folk Songs, published in four volumes between 1946 and 1950 by the State Historical Society of Missouri). Little else is known about the origin of the song." 
The Roud Folksong Index classifies this song as a vesion of Roud 4816, although it lists no printed or recorded versions of this song which pre-date Barnett and Rutherford's recording.
Overall, there are reasonable grounds for treating this composition as public domain, and there is no firm evidence yet to contradict this assumption.
Sweet William and Lady Margaret by Jean Ritchie recorded no later than 1961 (album released that year according to discogs): Spotify Album is called British Traditional Ballads so it is reasonable to treat this recording as public domain.
Willie Moore by Fred Starr recorded in 1958: Max Hunter Collection