There don't seem to be any traditional songs from Bideford that I can find reference to; nor any found being sung there. Truly local songs are rare in any case.

There was, however, a broadside printer operating there in the 1860s or thereabouts: J Wilson on Bridge Street. You can see a decent selection of his output at Bodleian Library Broadside Ballads:

Song sheets from J Wilson of Bideford

That will give you an idea of what sort of thing was circulating there at the time, at any rate.

Charles Kingsley included a verse of the Bonny Portmore / Sunderland for Me / Streams of Lovely Nancy type in his Westward Ho!, for which see thread Origins of Bonny Portmore

Apart from that apparent localisation, I can't think of anything. Perhaps other people will know of more.