Root wrote the tune we now generally think of as "Jesus loves the little children", but I'm not sure he has any connection with the tune called SHEFFIELD ("British Grenadiers"). FWIW there's a quite different tune called SHEFFIELD in one of the fasola tunebooks (not sure if I've seen it in the 1991 Sacred Harp or the 2012 Shenandoah Harmony. Will look it up.

I think the "British Grenadiers" tune is anonymous, and quite a bit too old to have been written by Root, though I suppose there's a chance he did an arrangement of it, quite possibly a hymnal arrangement, and perhaps he might have had a hand in introducing it into American hymnodic usage paired with the Montgomery "Hail to the Lord's Anointed" text...