Regarding the film's tune, I found the following in Middlesex Independent & W. London Star, October 14th, 1949 (p. 2):

Composer Brian Easdale, who won an Oscar for his "Red Shoes" film score, has written lyrical settings for three soprano songs sung by star Jennifer Jones in "Gone to Earth." These include a new setting for Shakespeare's "Sigh No More, Ladies" for unaccompanied voice; harp and choir setting for the traditional "Harps in Heaven"; and harp settings and words for "The Mountain Ash." Highlight of the score will be the theme tone poem, "Ride of the Black Huntsman," written for four harps, strings and horn quartet.

You can find Easdale's copyright card here, which claims "Three songs; from the film Gone to earth, music by Brian Easdale, [words by William Shakespeare, Brian Easdale and Marry Webb] 2/- (c) Chappell & Co., ltd., London; 11Oct50; EF0-5044.

On the other hand, here is Dransfield's recording; on his 2005 release Unruly (link) he claimed tune composition credits.

Though the aforementioned newspaper articled noted the song as traditional, I've found nothing to suggest the words have any origin other than Webb. Gladys Mary Coles, the author of Collected Prose and Poem: A Selection of Mary Webb's Hitherto Uncollected and Unpublished Work (1977) curiously included only the second and third verses (p. 120) in her compilation; Coles notes in the introduction that many of the poems and short stories were found scouring local literary journals and magazines, but doesn't point to specific sources for a number of the works. The overall message, however, is still that it was Webb's original work.

So I think my ultimate takeaway here is that Webb composed the lyrics herself, then Easdale and Dransfield both put it to (somewhat similar) musical settings using their own approximations of the original meter.