This is what the NHS says:
"You can take your baby swimming at any age, both before and after they have been vaccinated. It doesn't matter if they haven't yet completed their course of vaccinations.
Some leisure facilities may suggest that babies shouldn't go swimming until they have had all of their jabs. However, this advice probably dates back to when polio was common and people were concerned it could spread in busy places, such as public swimming pools.
There have been no cases of natural polio infection in the UK for over 20 years (the last case was in 1984). About 40 cases of polio have been reported in the UK since then, but these were thought to have been acquired abroad or occurred as a very rare side effect of the attenuated polio vaccine (containing the live virus) that was routinely used up until 2004.
The Department of Health says that you can take your baby swimming from a very young age. There is no need to wait until they have been vaccinated."
I take my baby swimming a lot. We go to Water Babies classes which we started when he was really little (I think 2 months?), they help because they give us things to practice outside of the lessons, I probably wouldn't have known what to do with him otherwise. He's 7 months now and can go under water very confidently and hold on to the side. They are a bit money grubbing though (the photos they try and sell you are outrageous) but there's no where cheaper doing a similar thing in our area unfortunately.
Alternatively, I have this book which looks quite good:
www.amazon.co.uk/Teach-Child-Usborne-Parents-Guides/dp/1409527700/ref=sr_1_1?ie=UTF8&qid=1429517619&sr=8-1&keywords=teach+your+child+to+swim
And as PP said, in most public pools you will probably need a wetsuit. DS is fine in a toddler pool now but as a newborn I would have still put him in one. In an adult pool I would definitely put him in one still, and will do for some time. We have this one and have been happy with it (our local pool is very cold, but he has always been fine):
www.amazon.co.uk/Splash-About-Baby-Neoprene-Wetsuit/dp/B00IK006OC/ref=sr_1_1?ie=UTF8&qid=1429517673&sr=8-1&keywords=splash+about+wetsuit+baby