Although, as Marje points out, (Farewell) Lovely Nancy (or whichever of many titles you choose for it) isn't related to Adieu Sweet Lovely Nancy, it's a nice song. There's a longer set, with tune, in the DT:  FARE YE WELL, LOVELY NANCY: in that case taken from Roy Palmer, Oxford Book of Sea Songs, 1986 (recently republished by Dave Herron as Boxing the Compass, 2001). The DT omits source information; that version was originally noted by Dr George Gardiner (text) and (probably) Charles Gamblin (tune) from George Lovett (born 1841) at Winchester, Hampshire. In January 1909, Ralph Vaughan Williams re-noted the melody because there was some doubt about the notation; it appears that he visited Mr Lovett and recorded his singing for later checking. Usually he gets credited as collector, but it's a bit more complicated than that.

The text Melani posted (a few years ago, now) is essentially Bert Lloyd's re-write of the Sharp set; the arch "reef your topsail" and the reference to Nancy's "behind" don't appear in any traditional or broadside versions that I've seen, and I expect that Lloyd made them up himself.

The Sharp version was noted, as Farewell My Dearest Nancy, from Mrs Susan Williams (73) at Haselbury Plucknett, Somerset, 26 December 1905. It has been a bit polished-up for publication, with a little additional material added. I don't know who changed Mrs Williams' "Dearest Nancy" to "Lovely Nancy"; Sharp himself published the song as Farewell Nancy with that bit unaltered.

There are a few broadside editions at  Bodleian Library Broadside Ballads:

The sailor's adieu.

Laws K14, Roud 527.