The Bodleian Library Broadside Collection  also has a number of (probably) slightly earlier copies:

Fall of leaves  Printed between 1796 and 1853 by J. Swindells of Manchester (2 copies showing significant textual variation from the above).
The fall of the leaf  Printed between 1797 and 1834 by G. Walker of Durham (5 copies, pretty much the same as the Harkness issues).

The Fall of the Leaf (printed by J. Hirst of Oldham in 1843) is an unrelated piece written by Thomas Rhodes.

Alfred Williams had a set from Thomas Dunn of Stratton St. Margaret, c.1914-16 (Folk Songs of the Upper Thames, 1923) and commented, The piece is old, and was a favourite throughout the Thames Valley.  Vaughan Williams got one in 1904 from J.W. Wright of Coombe Bissett, Wiltshire (Roy Palmer, English Country Songs, 1979; Palmer comments, The song appeared on broadsides in the nineteenth century under the title of The Fall of the Leaf, though it was considerably improved by the refining and polishing action of oral circulation)  Bob Copper gives a set from Rottingdean in Early To Rise (1976); I don't have time just now to see if he refers to the song in the narrative.  That's all I can add at the moment, beyond noting that such internet references as exist are mostly to unattributed texts or to recordings by revival performers.