I was going to suggest Wells and Shepton Mallet
They have some lovely surrounding villages like Croscombe, West/East Horrington, Oakhill/Binegar, Doulting, Chilcompton.
It's about 30 minutes to Street from Binegar [I used to go to/from Strode College as I lived nearby]. Bath RUH will probably be the closest big hospital which is about 30-45 minutes away. Though there are small local hospitals in Shepton and Paulton.
Wells has a lovely indoor pool, Shepton & Street have outdoor pools that are open in the summer and are lovely for a family trip
Wells has a small cinema, Yeovil, Bath and Bristol are your nearest chain cinemas.
Trains aren't too far away, Bristol Temple Meads has a train to Victoria in about an hour and that's 45 minute drive away, Bath and Castle Cary.
There is loads of lovely walking on the Mendip Hills, Weston Super Mare is about an hour away and is a lovely sandy beach. Though the Dorset coast is about an hour and a half away, so day trips to Weymouth and Lyme Regis are a treat.
Glastonbury has big price tags but IMHO isn't that special, Shepton Mallet had a bad rep 10-15 years ago but it's nicer now. The villages are not as cliquey as they used to be, most of the village schools do have a great rep. Binegar school has recently closed, but local schools are in Chilcompton and Oakhill [both are brill, me and DH attended them and our mum's still work in them].
DD is 13 months and there is lots going on locally for her to do [too much sometimes]
My experience of Wedmore was a little poor when I was a live in Nanny, not too much going on and the people were all quite mature and cliquey, but that was 12 years ago.
A friend hated growing up in South Peth, I think it was a trek to anywhere in particular.
Feel free to contact me if you want any further questions asked.