I have two dates in mind, 22 May 1940 and and 27 May 1940. It makes more sense that they started evacuating from the mole the 22 of may but I've also heard the 27 from BBC.
The 22 of May was the date when Capt. Tennant arrived to Dunkirk and he was, if I'm correct, the one who ordered the evacuation from the Dunkirk mole.
I don't think that the 27 makes any sense because this means that Capt. Tennant would have waited five days (from the 22) before he made the order and time was not really something that they had a lot of during the evacuation. Which is correct?