Swimming pools abound in Chiang Mai - pretty much every housing development has one and many of these pools also offer swimming lessons. The lessons we've seen are held almost exclusively in Thai and as the pools are also open to the public you will likely find them to be teeming with recreational swimmers, swimming teachers and adults and kids learning how to swim.
For a different way of teaching/learning, we found Baby Pool, which opened in late 2016. This is a dedicated swimming pool for teaching children from 4 months to 6 years old how to swim. The teachers are trained by Australian and UK swim associations and all classes are taught in English. Parents swim with their kids (although there are sessions for more advanced swimmers which don't require parents to get into the pool), classes are limited to 6 kids and their adults and emphasis is on safety & fun in the water. We used to take our 3 year old to very similar classes in London from 6 months, so for us, it's been nice to be able to do the same with our 7 month old here in Chiang Mai.
Baby Pool's facilities are fantastic and very clean. The reception area is air conditioned and has a cafe (unfortunately only sells ice cream and snacks, no food), tables and chairs (including high chairs) and a small play area. The changing facilities are spacious with big changing tables, lockers and shower rooms which include baby tubs and seats. The pool area has a big pool where the classes are held and a small 'holding area' where kids waiting for their class to start can wallow.
It may seem odd for a tropical country, but the pool is heated and indoor. The roof keeps the extremely harsh sun off, allowing lessons throughout the day, and the heated water is perfect for very young babies. During the cool season (Nov-Feb) the water in outdoor pools gets quite cold and even in April, one of the hottest months, the water can still be a little cold for the very little ones, so we've found this pool to be ideal. The only issue is that the set up (basically a big pool of warm, still water in an enclosed area) is a haven for mosquitoes. Amusingly, the managers have dealt with this by arming the cleaners with electric racquets to zap the little critters.
Weekly lessons last 30 minutes and are run in blocks of 10 weeks. It's not cheap - prices are similar to London prices - but we've been very happy so far with the lessons and our kids always enjoy themselves. A good reflection is the progress we've noticed in both kids in the 6 weeks we've been having lessons for, and so we've signed up for the next quarter.