Purely local songs can be hard to find (though they do exist), but the general Derbyshire repertoire is easier to get a handle on. You might be interested in Veteran's recording of the Derbyshire traditional singer George Fradley, for example, which can be had by mail order from  Veteran:

Veteran Tapes VT114: One of The Best: Songs from Derbyshire. George Fradley.

He had a range of songs from the comic to the classic, including versions of Long Lankin and The Cruel Sister.