I'm also from Yorkshire, and this was a word my mother used too. I think the word you're looking for is 'chavel' rather then 'chevel'. The meaning is definitely the same - chewing, gnawing away at something. My mother always used it when the dog was grooming herself - 'Stop chavelling!'.
According to Merriam-Webster it means 'nibble, gnaw' and comes from the Middle English 'chavlen, chaulen', and Old English 'ceafl' (meaning 'cheek or jowl').
Definitions says that 'to chavel' means 'to chew'. The same site says that it's also a noun, meaning:
The jaw, especially, the jaw of a beast
I found a very interesting PhD thesis from 1952 by Albert Lyon Hoy, which gives an etymological glossary of the East Yorkshire dialect. This link will open a PDF of his thesis: 'chavel' is on pg 93.
Loy found a couple of examples of the word in Old/Middle English texts, in The Owl and the Nightingale, line 284 and in Ancrene Wisse, line 70.
Loy also found a definition in an old publication:
'Chawle, to chew imperfectly' (from A Glossary of Words Used in
Holderness in the East-Riding of Yorkshire, published by the English
Dialect Society in 1977)