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Filton is a good one, actually warm for babies as well
Look up Dinky Dolphins in Filton - not been myself but apparently it’s a drop in session with instructors around and £5?
Hengrove is probably your closest, or Bristol south pool does casual swim on Fridays where the pool is warmer. Hengrove can be a bit on the cooler side so I suggest getting a full swim suit (splash about) you can get them on vinted cheap.
Hengrove leisure centre
Horfield Leisure Centre - Tip - download the Everyone Active app to book as the website is atrocious. Baby pool = teaching pool. Not at all obvious but the receptionist explained.