Well, I live within a 45-minute drive of both Stroud and Frome, just outside Bath, and used to work in Frome. I wouldn't live there. It's miles from anything, which leads to an insular feel about it. The Asda does have cheap fuel, though.
Corsham or Bradford-on-Avon? Cotswolds feel about them, but less out of the way. Only a 20-minute drive into Bath, too. Lots of lovely walks, particularly in BOA, where you can walk along the canal and take the DC to the canal side playgrounds for a paddle in the river, or to explore the Tithe Barn (nothing in there but a source of fascination for mine when they were small).
I know Corsham schools better than BOA ones, but I have friends in BOA who have no complaints. Corsham ones are good, too.
I grew up in London and moved to Bath as an adult. You might find that going completely to the country is a bit jarring for you when you've been used to having everything on your doorstep. Somewhere with easy access to the motorway or other towns or a mainline station will make the transition easier. We've moved out of Bath now but are still nearby, in a town with great schools and a fab community.
The above suggestion of Keynsham is a good shout, though I'm not sure the town centre has the pretty stone buildings vibe you seem to like. Good amenities, though, easy access to Bristol for your city fix and the Chew Valley for your country one.
Consider whether you want to be your children's taxi as they get older. Especially in places where the bus is once every two hours on a Sunday. For us, it was important to live somewhere where they could become more independent. We have great transport links where we are, so the DC can spread their wings a bit as they get older. DH and I both grew up in cities, so this was important to us.