Gosh. Just found this. Thank you to those pointing out my thesis.

SONGS: Yes do download my thesis on Cumbrian folk songs and have a read. Link here:
https://www.academia.edu/32901994/FOLK_SONG_IN_CUMBRIA_A_DISTINCTIVE_REGIONAL_REPERTOIRE
the appendix lists

Or perhaps Mary Emmet's thesis on hunting songs:
https://etheses.whiterose.ac.uk/29320/

In my thesis, by all means just skip to conclusions, if you must, but do take a look at the Appendix, which lists some 500 songs.
There are some Border Ballads there, some set in Carlisle and environs, but do bear in mind that these are very much a literary tradition, and almost all edited or re-written by Sir Walter Scott. For recordings, yes, there are old Paul and Linda Adams recordings and some other Fellside recordings, more recent recordings by Striding Edge, Steve Wharton and Dave Camlin, but also seek out 'Pass the Jug Round' - recordings made in two pubs in the Carlisle area in 1953 (download available from Veteran)- and check out the British Library's Sound Library, where you'll certainly find some hunting songs. Peter Kennedy also recorded in Cumberland and Westmorland in 1958/9.

TUNES: And when it comes to Lakeland tunes - well, there are more than anyone could get to grips with in a lifetime! Commercial recordings by The Boat Band, Striding Edge and the Lakeland Fiddler, and many, many transcriptions (score, ABC and midi) of the manuscripts of the many late eighteenth and early to mid nineteenth century Lakeland fiddlers (and flautists) are given on the Village Music website. https://www.village-music-project.org.uk/?page_id=84

DANCES: many of the country's clog/step dance teams perform Lakeland clog and step dances. And many ceilidhs feature eg Cumberland Square Eight, Cumberland Reel, Westmorland Cottagers etc as social dances.
No Cumbrian folk music? You must be joking!!