I started bringing my sons in the swimming pools at 4 months, but they could really swim launching them underwater around 7 months. At that age, children have still a very strong mammalian reflex, so they cannot drown, keep their eyes open, open the mouth (without inhaling) and swim some sort o of "dog style".
Around one year, I gave them fins, mask, snorkel and small inflatables on the arms.
With that equipment they did snorkel around easily, also in the sea. We had nice holidays in Crete and in Sardinia when our children were that age. They do not fear swimming in deep water, even crystal clear, and they love watching fishes from the surface.
Both were swimming with fins and mask well before walking.
Around 18 months we started training them free diving. In the swimming pool you launch toys on the bottom, and they have to catch them. First thing they need to learn is to equalize, and around 18 months they can learn it. Maximum depth was 120cm, indeed: so they cannot touch the bottom, but the parent in the water can, providing perfect safety.
After learning free diving, they learn snorkeling WITHOUT inflatables on their arms. In the sea, a shorty wet suit helps providing some additional floatancy, and cold protection, still allowing them to free diving for catching shells, sea stars, etc.
Around 2 years it was time of their first scuba tank. It was a small 2-liters alu tank, back mounted, with a single reg and no SPG. In the swimming pool (so max depth 1.2m), equalization already mastered, they had to learn to always exhale when surfacing, either with the reg in their mouth and without.
Around 3 years they did master perfectly removal of the regulator and getting in contact with it even if flooded, removing the mask and evacuating it, and buddy breathing with me with a single reg.
Also the kicking style was now almost perfect, and I purchased them their first pair of serious free-diving fins, providing proper thrust for emergencies. At that that pint they were ready for the sea, and they had they firs dive at 4-5 meters depth in the sea.
In the following years the progress was slow but constant. Around 7-8 years they were already able to reach the maximum depth allowed for children (10m), to pass below arches and small caverns, and to orient themselves coming back to the entry point without help. At that age they were using a 5-liters steel, back mounted with backpack but no wing (proper weighting i mandatory without BCD).
We introduced the BCD only after 10 years, and at 12 years both have been certified Junior Advanced OW.
With children you must start very early, going to the pool at least one per week all around the year, spent at least one month at the sea each summer, and proceed very, very slowly. Any hurry in reaching a goal can be risky, you must give each child the time he requires.
The time spent underwater with my sons and my wife (who also is an instructor, and she is specifically trained for children below 3 years) was probably the best time of my life.