I think Bronson said that Rimbault was responsible for the editorial changes and tidying up. I haven't got a copy of Rimbault so I can't be definite, but that seems the likely scenario. If Rimbault was fairly slack about his source, then Chappell would just assume the tune was traditional in general.

Chappell published an earlier version of PMOT in 1840, titled 'A Collection of National English Airs consisting of Ancient Song, Ballad & Dance tunes'. It contains pretty much the same material as the later PMOT but there is no 'Little Musgrave' which again suggests he got it from Rimbault already altered as opposed to Motherwell.