Dick Greenhaus posted the DT text and would be the person to ask. He credited no source at the time, though. It seems that MacColl & Seeger recorded a set with the same first verse (Long Harvest, volume 6, 1967), and the text appears on a number of websites. None of them, sadly, bothers to say where they found it. Perhaps MacColl did his source the courtesy of a credit.

Lesley Nelson's set came from The Colonial and Revolution Songbook (Keith & Rusty McNeil); unfortunately, that book is probably worthless as a source. In the "Colonial America 17th Century" section, along with The Willow Tree, they include The Great Silkie Of Shule Skerry (sic), which has never been found in tradition outside the West of Scotland; and even there, not prior to the late 19th century.

The compilers, if the contents list ( http://www.mcneilmusic.com/revsngbk.html ) is anything to go by, seem to have been almost stunningly ignorant. Sadly, once a lie appears in print, people quickly start to believe that it must be true. People like the McNeils ought to be thoroughly ashamed of the disgraceful misinformation they have spread about.