I also have a video called Waterbabies by Little Dippers UK. It's a 21 week by week instruction and demonstration of how you can teach your baby to swim. Their main aim is to teach babies to swim to the edge or to turn and float in case they do fall into the water.
They also say that most drowning occurs when the baby panics and takes in water, but if they manage to stay calm, it actually reduces the chances of drowning.
In the video, they have around 10-15 babies doing the demo and the youngest is only 5 weeks old. It's really amazing to watch them all being taught how to swim and also when they actually swim.
However, at the end of the day, it is the practical experience which matters so I definitely still think a real coach will be better than any videos or books.