The fact that Pete Seeger referred to his set of the song as Another song from England, collected by Cecil Sharp many years ago and titled by him "Waillie, Waillie" would tend to confirm (in spite of the odd spelling, which however also occurs in Silber and Sandburg) that it was the set published in the U.S.A. in Sharp's One Hundred English Folksongs (Oliver Ditson, 1916), though with a verse added from somewhere or other. The attribution to Sharp's English Folk-songs from the Southern Appalachians is certainly an error.
The published set was, as I mentioned, a collation; so far as I can tell, drawing on three sets all noted in Somerset. I don't see any evidence of interpolations from other sources. Working from the material printed in Cecil Sharp's Collection of English Folk Songs (ed. Maud Kapeles, 1974: not a full edition of the collection, but the most extensive so far published), I'd break it down as follows:
The tune came from Caroline Cox, as did verse 2 (very slightly modified with reference to a set noted from James Thomas at Cannington in 1906); verse 3; verse 4 (also slightly amended from James Thomas' set); and verse 7 (with just a few words modified for the sake, I expect, of euphony).
Verse 1 was from Elizabeth Mogg (mentioned above) slightly edited. Verses 3 and 6 are from James Thomas (slightly edited); verse 8 is from Elizabeth Mogg, again slightly modified.
This collated form of the song has been known in America (in theory) since its publication there in 1916. Of course, there are plenty of cases of songs being taken up from such printed sources by singers who would in the general sense be considered "traditional", but the only possible example that I'm aware of in this particular case is Almeda Riddle; and I have no details about that, beyond the fact that she begins with the Water is wide verse, which would tend to suggest Sharp's collation as her (ultimate) source.
I don't recall where I read that Pete Seeger had the song from Peggy, but it seems likely enough. I haven't seen his book, so I don't know what the extra verse he added might have been; if it was the silver bells one that turns up in a number of Revival recordings, it could have come from any number of places, being a not uncommon "floater". At all events, the source was evidently a printed one judging by Pete's notes; if not Sharp's book, then another in which the collation had been reprinted.
That's about as much as I can manage by way of clarification at the moment. The unpublished Appalachian fragment I referred to is in Sharp's MSS (Folk Words p.2544 / Folk Tunes p.3456; noted from Mrs. Jane Gentry of Hot Springs, Carolina, 14th September 1916, and beginning As I walked out one morning in May [!]); I only know about it from the reference in the Roud Folk Song Index.