Steve Roud's Broadside Index includes the copies of the Madden collection at the Vaughan Williams Memorial Library, and has what appears to be this one as Farewell Ye Green Fields (first line, Farewell Ye Green Fields and Sweet Groves). Under that heading, a copy can be seen at Bodleian Library Broadside Ballads:

("Farewell ye green fields, and sweet groves ...")  Printer and date unknown; "Sung by Mr. Vernon, at Vauxhall". On the face of it, there doesn't seem to be any relationship to Green Fields of America.