My son finally, alhamdulillah, was able to do a torpedo (swim with arms in a torpedo position and legs kicking). It took 6 of one week long lessons for him to reach this stage over a period of 2 years.
Taking to the waters varies between each child but from what I have noticed it usually takes a long time. With my other son it took him 6 lessons as well but over a period of 5 years. At first they just floundered and splashed around. But once they have gained a water confidence, they become more relaxed. This is the point where they realised that they will not drown if they relax and 'swim'. It was the fear that made them tensed up and thrashed about.
If you are teaching your child to swim then teach them to overcome this fear because there is no point in teaching technique when they still have this fear. Having watched my children taking lessons I have also noticed many other children doing the same thing through fear. It's a similar pattern, once they overcome the fear you can see a marked difference - they become calmer.