Hi there. The university has just closed its pool to the public for safety reasons. I think the North Parade pool is fine for families, there is a small shallow warmer pool adjacent to the main one for younger children and a flume for the older ones! There is also a pool at Culverhay school which is open to the public. I believe they all have activities for children.
Culverhay is quite a deep pool, probably better for those that are more confident swimmers. South Wansdyke Pool, in MSN has a nice shallow stepped bit, it's not as warm as some of the other pools. We are making the trip to Wells for the parent/toddler sessions, it's nice and warm, shallow and reasonably priced, it is more your own thing, rather than an instructed session, which suits us, there is an instructor you can ask for advice.