That would make the "tune" Rose tree and the "song" Bonny Bunch of Roses. Sorry to be picky but I have this problem all the time with people using the wrong titles for songs/tunes. Guess it comes with being a Virgo.
I am listening to the show now and the tune is definitely Rose Tree.

A few tracks further down the list they credit:

Trad: Song and tune (as yet unidentified)
Val's Sunday session at the Kelham Island Tavern in Sheffield
BBC Field Recording

The song is definitely Dido Bendigo, a Yorkshire Foxhunting song.
It sounds to me like Gavin Davenport of the wonderful English group Crucible who is singing the lead. I also think I could hear Jess Arrowsmith singing in the chorus. The group is from the Sheffield area. I will email them and get them to verify if it was them.
JohnB