"The Nobleman and the Thresher" indeed is in the DT, but must be found under "The Nobleman....", not under "Thresher(man)."
It does not seem to be in "A Traditional Ballad Index," or if it is, under a name other than Thresher(man) or Noble(man).

The version collected by Sharp is listed in library.efdss.org. and the song is listed as "The Jolly Thresher" in the folk music index. It also is in Sam Henry.

I could not find a copy at the Bodleian under "Noble(man)" or "Thresher(man). It should be there, but what title?

Link to an early (extended) copy would be appreciated.