Masato, that commentary by Pullen helps a lot. The melody used by Bach in his Cantata 212 (Mir hahn en neue Oberkeet = Peasant Cantata) could be the same as the one used for Farewell ye green fields and groves. It is likely that Bach based his melody on a popular tune.
Lewis Edson then used the same tune to set Newton's "How Tedious and Tasteless," as previously mentioned.

Now if only someone will come up with the tune from Baring-Gould's Songs of the West." (Then both of us will say WOW! WOW!, WYSIWYG). A reference to the American uses of the tune would also be helpful. Not in Randolph or Brown, at least with titles Greenfields or Farewell ye...