The early 1500s.
From Pottermore's article on the Leaky Cauldron (thanks to @Skooba for finding this!):
The Leaky Cauldron was there long before Charing Cross Road was even planned; its true address is number one, Diagon Alley, and it is believed to have been built some time in the early 1500s, along with the rest of the wizarding street.
(based on less canonical information):
According to the third W.O.M.B.A.T. (Wizards' Ordinary Magic and Basic Aptitude Test), it's possible (though not confirmed, this being one item in a multiple choice test) that:
The oldest building in Diagon Alley is Gringotts Bank; the other shops grew up around it.
According to the Harry Potter Wikia and other unreliable sources, Gringotts was established in 1474 by a goblin called Gringott. This would set the formation of Diagon Alley at 1474.
Unfortunately, all of the above is based on dodgy and unreliable information. The best I've found canonically is an upper bound of the early 1500s. Daisy Dodderidge (1467-1555) built the Leaky Cauldron as a gateway to Diagon Alley, so the street must have existed by that time.