Hi originally from Wiltshire here but I did attend King Edwards in Bath so feel that I also have a stake in Bath! Lots of friends lived in Bath so feel I know the city reasonably.
Bath is a great place to live especially for young people. Its easy to walk around the city, it has good shops, the schools are good and there is a good range of stuff to do. Lots of nice parks and you can live a life not too afffected by the tourism. Its also reasonably well connected - its easy to get to Bristol by train and London is easy to do. If I didnt come from the area I would have definitely considered living there with my family but I think I would want to be within 2-3 ,miles of the city centre to benefit fully from city life.
Of my friends from the Bath area loads lived up on Lansdown or in Weston - they walked to school or caught buses. Several also lived in Bathampton which is a quick journey in or Batheaston which is a slightly longer journey and can get congested. There were also lots in the Prior Park, Combe Down, Odd Down areas on the outskirts of the city. In terms of near places several people came from Freshford Box, Saltford, Keynsham etc which are all worth checking out
In terms of travel to King Edwards the area from which people come is pretty large: there are lots of coaches from places like Trowbridge, Frome, Bradford on Avon, Westbury (and villages in between) which serve KES and Royal High - taking people directly to school.
There is also train travel - I was one of the 30-40 KES and RHS pupils that got the train from Chipppenham every day. Its only a 10 minute journey and the train at 7:45ish meant I got to school for 8:20 ish.
Its a 1 mile or so walk from the train station to KES taking in Pulteney Bridge and Pulteney Street which is a glorious walk to school! There is also a bus.
Of my friends who caught the train several lived in villages such as Kington Langley, towns such as Malmesbury and Chippenham as well as various villages along the A4 to Calne etc. Check out these places for the Wiltshire rural life. These have the advantage of being better located for the A4 and M4 for travel for London if this is important and you get more for your money. But you probably will end up travelling to Bath for culture, hospitals, shops etc. Also small Wiltshie towns such as Trowbridge can feel a bit like backwaters if you're used to city life.
Public transport is not very comprehensive outside Bath and so you will probably find yourself doing quite a bit of driving for your children. Also not being in Bath meant that everything became more complicated when arranging to meet up with friends etc so I would recommend living closer to Bath if possible, and certainly on a bus/train route if you want your chilfdren to be indendent!