Sheet music at American Memory has sheet music for "Beneath the Willow Tree," "Words by Thomas Dibdin, Music by J. B. Herbert," 1884, Balmer and Weber, St. Louis.
It is the version beginning 'Oh! take me to your arms my love,/ For keen the wind doth blow,...' (See version from the Bodleian, posted 24Sept08, 0326PM), but with changes.

I have the book by T. Dibdin, "Songs by Charles Dibdin, with a Memoir," 1875, which compiles the songs by Thomas and C. Dibdin Jr. as well as Charles Dibdin; "The Weeping Willow" song is not included, so I don't think it is a song by any of the Dibdins.

I left out the last four lines of the first verse of the version posted 24Sept08, 0326PM. Here is the verse:

THE WILLOW TREE

1
O take me to your arms, love,
For keen the wind does blow,
O take me to your arms, love,
For bitter is my woe;
She hears me not, she cares not,
Nor will she list to me,
Whilst here I lie alone to die,
Beneath the willow tree.

Sorry, mayhap a clone can correct my mistake.