There is a large community of people that post on the internet a lot of video files with mermaid tails taken to the pool for a swim. Some of these videos can be found here.
As you can see in the video, the girls wearing the mermaid tails can swim very well under water. The first thing that pop into my mind when I tried to imagine how will it be to wear such swimsuit in water was that the feet must be moved in tandem and this could be very hard since the classical way of swimming is to move you feet independently.
The tail has a fin at the end, made out of hard but flexible plastic, covered in soft scuba diving material, that allow the swimmer to introduce his feet and dress over an elastic material that resembles a fish tail.
I must say that the lack of experience in swimming with this mermaid tail must be compensated by the use of inflated armbands as a safety measure.