Roud has no record of a song with the New Britain title.

Turner's book on Amazing Grace has this to say on the origins of the tune has this to say (about the Columbia Harmony):

"Two of the tunes, “Gallaher” and “St. Mary’s,” proved to be not so unfamiliar. They both shared striking similarities to the tune we now associate with “Amazing Grace.” “St. Mary’s” accompanied the text of another Isaac Watts hymn, “Arise, My Soul, My Gentle Powers,” and “Gallaher” accompanied the Charles Wesley hymn “Come Let Us Join Our Friends Above.”...

Spilman and Shaw found most of the tunes in Columbian Harmony in other tune books. Marion Hatchett has been able to establish the origins of all but 12 of the 159 tunes, the majority coming from shape-note collections published during the previous sixteen years. Unfortunately for researchers of “Amazing Grace,” “Gallaher” and “St.Mary’s” were two of the 12 that couldn’t be sourced...

The fact that such a high proportion of the tunes were from other books suggests that neither Spilman nor Shaw was a composer. There is no mention of writing or playing an instrument in Spilman’s autobiography. This leaves two main possibilities. The two tunes could have been lifted from a book, all copies of which have since vanished. Or they could have been variations on a single tune that was popular as a folk melody but that had never previously been transcribed."

And of Walker's use of the tune:
Among the tunes that had already been published was the same one that had been collected in two early forms by Spilman and Shaw, then shortly afterward named “Harmony Grove” by Carrell and Clayton. Walker polished it up, named it “New Britain,” and paired it with the words of “Amazing Grace.”

This last seems to suggest that there never was a song called New Britan. (Turner discusses people's suggested origins for the tunes but (I think) concluded that there is no evidence for origin).

Mick