Following up on this thread, I returned to the US yesterday evening after a delightful 3-week visit. Plans changed a bit. As expected, I did spend a week bicycling in the Cotswolds followed by a week in eastern Somerset. I didn't visit Wales. Instead, I spent an additional week noodling around Wessex.
I'd been aware of the network of (barely) single-lane roads, but on previous visits, I'd never experienced them. For bicycling, I found them ideal. Motor traffic was sparse, and what I encountered was skilled and courteous. Yes, I'm aware that's not universal, but it was my experience. Returning the favor, when my planned route led me to busier roads on which I was uncomfortable, I changed my route and went elsewhere.
Surprisingly, weather smiled on me. Mostly, it rained overnight, and on the one day with torrential downpours (last Saturday 19 July), I availed myself of an amenity you've sited at convenient intervals. You call them "pubs". When I saw cloud-to-ground lightning directly in my path, and counted the time between flash and crash -- it was about a mile ahead of me -- Somerset Arms in Maiden Bradley offered a prudent alternative and a friendly welcome. Another lunch stop worth mentioning is the Mells Café. As with the Somerset Arms, it's not the food or drink that was so outstanding though it was good; it was the welcome. And the Mells Café had a floor pump in the corner. No, I didn't need the floor pump, but these folks give more than lip service to opening their door to bicyclists.
Summarizing: I couldn't imagine a better way to enjoy southwest England. Thanks for the welcome.