In his treatise "De re militari" ("On military matters"), writen by Vegetius in the 5th century, Vegetius advices the reader to have his soldiers learn to swim. It's a bit unclear whether he described the actual training, or a hypothetical ideal training. However, he states that it would be preferable to teach every soldier how to swim.
The book was widely influencial and often read in the Medieval and Renaissance, some military leaders even tried to adapt the lessions therein.
Edit: I forgot to mention that he claims that "the ancient Romans" would have taught it to their young soldiers.
In 1918 the Austro-Hungarian battleship SMS Szent István was torpedoed by Italian boats, and it sunk within 3 hours. However, of the crew of 1087 only 89 died. The reason was that that in the Austro-Hungarian Navy it had became mandatory for sailors to learn to swim... in 1894. I don't know the answer to your question, but I think it shows that it's quite a recent thing to teach people how to swim.
While it would be hard to generalize across all societies and cultures, especially considering the role water would play for different regions inside the same empire, there are historical records of people swimming from Ancient Egypt, Greece, and Rome.
As far as the standardized teaching of it, this did occur in the 14th century as knights were drowning in their armor. And in the 1600s Japanese Emperor Go-Yoozei did institute a swimming program for schoolchildren. This is a timeline of events I found, although it would need to be properly vetted.
Wikipedia also has a page dedicated to the subject.
Your question implies that the teaching of swimming is a very recent phenomenon, and that is not the case. Since it became a sport, including an Olympic sport, the teaching of swimming would be much more about socio-economic and cultural reasons than anything else. The history of swimming as a competitive sport dates back to at least the 18th century in England and other countries in Europe, and it was included in the "first" Olympic games in 1896.
The Talmud lists teaching children how to swim as one of the potential obligations of a parents towards their children.
האב חייב בבנו למולו, ולפדותו, וללמדו תורה, ולהשיאו אשה, וללמדו אומנות; וי"א: אף להשיטו במים
The father is obligated regarding his son in his circumcision, in his redeption, in teaching him Torah, in finding him a wife, and in teaching him a trade. And some say: in teaching him to swim in water.
Babylonian Talmud, Kiddushin 29a. Note that the text earlier makes clear that mothers are obligated in these things, too.
That said, there's no good way of knowing how widely this was followed.
Sorry, but this question has been removed for being a throughout history question. It's really impossible to answer this in any way besides giving a list of trivia bits about swimming in different cultures, which is the reason we have this rule.
You may want to suggest swimming as a Tuesday Trivia feature, or ask in the Friday Free-for-all later today.