Bartok used a Transylvanian relative of the tune in his First Rhapsody for Violin (and piano or orchestra, there are different versions). That was 20 years before Copland used it, and it obviously isn't directly derived from the Shaker tune. Both are two-part tunes but only the first parts match.

I presume there are commentaries on the Bartok that identify exactly where he got it (he didn't say at the time) and place it in a Hungarian tune family.

Here it is in a YouTube perfomance - 4'38" in:
Erno Kallai plays Bartok Rhapsody no. 1

My guess is that both derive from a common Central European original.