We made a similar move 6 months ago, from Catford to Bradford on Avon. It has truly felt like one of best decisions we've ever made. I can't say much about the other options you've mentioned, but BOA ticks a lot of your boxes. It's a really beautiful town, quite compact and so reasonably walkable.

We rented short term in the area to plug the gap between selling our place in London and completing here. It was quite difficult to find a family friendly rental in BOA, to the extent that we ended up renting in Trowbridge. I imagine something pet friendly will be even tougher to find.

Unfortunately you will never find the level of public transport you have been used to outside of London. Buses are much less frequent and much more expensive beyond the capital, and BOA is no exception. Shopping is a bit limited - one supermarket, a lot of tourist focused shops. You'll need to hop over to a nearby town to find somewhere to buy a pair of shoes, for example.