You'll be lucky if any of them do!

My parents live in South Devon, and coverage there is
very patchy.
In any case, the answer to this will vary mile-by-mile - it's highly unlikely that any one provider has better coverage right the way across an entire county.
Assuming you know the postcode of where you're going to be, check out this page:
sitefinder
That will give you a map of mobile phone masts in the area. (I'm not sure how up to date that is, mind you).
Each of the individual providers will also give you a map with estimated coverage in any given area. Try this page for links to all of the individual providers:
Direct Mobile Phones : Check coverage maps for Orange O2 Vodafone Three 3 T-Mobile mobile network coverage in your area.