I took my youngest to a Mums & Tots session at the local pool from the age of six months. It was all about playing games and gaining confidence in the water. As he got older, he started to learn some basic self-survival, such as how to catch and hang on to various floating objects. It gave him lots of confidence and he was swimming unaided by three and a half (although he refused to take off his deflated armbands until his 4th birthday).
It meant we were able to take him canoeing and sailing from an early age without the fear of him panicking or being silly if he did happen to fall in (which never happened). At 7 he wanted more swimming, so I sent him for extra swimming lessons and he joined a swimming club when he was 10, competing at county level. At 12, he was out-swimming most of the 15 year olds at his school.
Over and above the swimming, he also developed a certain maturity of attitude and a quiet self-confidence from being with the older boys and girls around him. I think it's part of the reason he's never been bullied (his brother didn't start swimming until nearly 2, hated the water and all the camaraderie that went with it and had a lot of bullyng problems).
I say go for it with your daughter, possibly via a parent/toddler group where you'd both get some support, rather than formal lessons. I know from doing hydrotherapy and swimming with learning disabled teenagers what a difference it can make. It improves and builds confidence, co-ordination and general health. When water takes the body's weight, for instance, it gives the limbs a chance to practice what they should be practicing without worrying about holding the body up.
Your health visitor may be even be able to recommend some groups or clubs you could join, with instructors who are experienced in teaching children like your daughter.
Good luck.