Hi, I had to wait until my twins (now 4.5yrs) were nearly 3 to have swimming lessons. One was fine but the other was a screamer.
Actually, she still is sometimes. We were actually thinking of pulling her out for a while if we couldn't get her to calm down a bit because she started getting upset even the night before her lesson. So, I spoke to the teacher and we are taking things very gently and not pushing her at all. Once she was assured that she wont be forced to do anything that made her scared she was happy and even smiled.
If I take her in the water myself she will scream and cling even worse than what she does with the swimming teacher. What I did was speak the the people in the office about her so that they could put her with the best teacher for her. Some children can do well with the tougher approach but others need a more gentle encouragement. It just depends on the child.
DD1 was a screamer too. She can swim now but still doesn't have the confidence to swim very far. But if you could have seen her when she first started you would see she has made alot of progress.
IMO the teacher they get put with can really make a huge difference. The evidence was the other week when DD3 who is NOT scared at all and will happily do huge bomb style jumps off the diving block all by herself. Her teacher was away sick and the replacement was a BOY

. She took one look and ran away screaming! They found another teacher for her and she was happy with the girl. In her opinion swimming teachers need to be girls.