Parallel Anthology Track 17
Track 17: The Carter Family - John Hardy was a Desperate Little Man Victor 40190-A; Recorded in Camden, New Jersey, 10.5.1928
According to wikipedia, A P Carter died in 1960.
The Roud Folksong Index classifies this song as a version of Roud 3262. A Roud number search for this tune returns a number of versions collected prior to the Cater Family's recording, dating back to around 1900 and from various states (N Carolina, Kentucky, W Virginia).
According to the Where Dead Voices Gather blog, "The real John Hardy was an African-American working in the railroad tunnels of West Virginia. He murdered a co-worker during a crap game and was sentenced to hang on January 19, 1894. [...] Alan Lomax provides the following additional information: [...] Hardy was tried during the July term of the McDowell County Criminal Court, found guilty and sentenced to be hanged. While awaiting execution in jail, he is said to have composed this ballad, which he later sang on the scaffold. [...] His ballad appears to have been based upon certain formulae stanzas from the Anglo-Saxon ballad stock.... [...] Whether actually composed by Hardy or not, 'John Hardy' became a popular song and has since been performed by numerous artists. It was first recorded in 1924 by Eva Davis."
This article:  from 1919 claims that this song had been in existence since the 1890s, and that there were several versions. Many of the lyrics of the Anthology version can be found among the various versions given in the article. Although on the Anthology, this track and Track 18, "Gonna Die with a Hammer in My Hand", are separate songs, the 1919 article discusses the possiblity that John Hardy (famous for a murder) and John Henry (famous for his ultimately fatal competition with a machine) were in fact the same person. Some lyrics from Track 18 can be seen in one of the versions of "John Hardy" given in the 1919 article.
According to the Old Weird America blog, "'John Hardy' stands right next to 'John Henry' as one of the most popular 'figure' in the folk song tradition (In the Anthology too, they are next to each other). In fact, many people combined the two songs and many scholars confused the two characters as Alan Lomax once said. Both were black railroad workers but their story is quite different. The historical John Hardy killed a man during a crap game and was hanged for his crime. Before his execution he wanted to make peace with God so they sent a preacher and went to the river to baptise him. On the scaffold he claimed his repentance for his crime and probably sang some verses that would be included in the ballad that bore his name. The origin of the song itself is hard to determinate as it can be a mix of spontaneous verses of work songs and white balladry put around the story of John Hardy’s life. But, like many other songs, it was sang by black folks before whites began to sing it. And now, except for the famous Leadbelly rendition of the song and maybe a few other, all the recordings of the song that i heard were by white people."
Given all of the above, there are very good grounds indeed for treating this composition as public domain, but the appropriation of copyright by A P Carter would mean that it is in copyright until 1st January 2031.
John Hardy by Pete Seeger album released in 1957 (according to www.folkways.si.edu): Spotify Liner notes treat this song as traditional.
John Hardy by Paul Clayton album released in 1962 (according to www.folkways.si.edu: Spotify Liner notes make no claim to authorship or arrangement credit and even imply that the song may have been written by John Hardy himself
John Hardy by Lonnie Donegan recorded in 1956-62 (as this is taken from album The Collection 1956-1962): Spotify No information found on whether Donegan claimed any arrangement credit for this version
John Henry by Odis Bird recorded in 1958: Max Hunter Collection Field recording