Here's some additional information I found.

"In an unpublished study of this tune, Marion Hatchett (of the School of Theology, University of the South, Sewanee, Tennessee) provided reproductions of no less than seventeen versions. ... This tune was known not only in a variety of versions but also by a variety of names, and it was associated with a number of texts other than "Amazing Grace" in its early history--Hatchett associated it with seventeen different texts. William Walker, who called this tune NEW BRITAIN, was the first to publish it as a setting for "Amazing Grace" in his Southern Harmony of 1835. Among the texts associated with this tune in Hatchett's study, only Johnson set it to "Alas! And Did My Savior Bleed."" (Andrew W. Johnson's version was published in several books in the 1830's & 1840's with the tune name 'Symphony')

Eskew, Harry. "Andrew W. Johnson's The Eclectic Harmony: A Middle Tunebook in Middle Tennessee." in Notes v. 58 no. 2 (Dec. 2001) p. 291-301.