The First English Manual on Swimming Was Published in 1587

We should clarify that Everard Digby's book De Arte Natandi was English in the sense that it was published in England. Digby, a theologian at Cambridge University, wrote The Art of Swimming in Latin. A few years later, Christopher Middleton translated it into English, which you can read online here.
The Public Domain Review describes De Arte Natandi within the context of European swimming practices of the time. The crawl was seen as an uncivilized stroke, but Digby does provide practical help for people who wish to swim through other means, such as the sidestroke:

This kinde of swimming, though it be more laborious, yet is it swifter then any of the rest, for that lying vpon one side, striking with your feete as when you swimme on your bellie, but that the pulling in and thrusting out of his hand, which then did onely keepe him vp, doe now helpe to put him forward: for onely the lower hand supporteth his bodie, and the vpper hand roweth like an Dare, as in this example.

De Arte Natandi came with at least thirty illustrations, all of which evidence that the swimsuit is a rather recent invention.