If Atwater-Donnelly said where they got the song, we may be able to help. Obviously, traditional songs are often sung to a whole range of different tunes, so guessing probably won't help much.

The list of tracks on the recording ( http://www.atwater-donnelly.com/music-poetry.htm#Culled-Garden ) names mostly songs that were very well-known standards (to the point of being hackneyed) at the time. Provided they acknowledged a traditional source, as the writeup on their website would lead us to expect that they would have done, then it shouldn't be too hard to trace.