My unabridged dictionary says that it's ch as in chair, both for chevy and for Cheviot. It notes that an "sh" sound in either case is an American pronunciation.

The first pronunciation it give for Cheviot calls for a short e. That would jibe with the "y" in Chyviot, above. This would make it a short vowel of some sort, as in "Ynglonde."

If you ask why Americans are saying Cheviot often enough to make it into a dictionary, it's because Cheviot is a breed of sheep as well as a range of hills.

I'm sure that Americans have been influenced by the word "Chevrolet."
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Thanks for bringing up the wonderful world of Middle English.