score:7
Encyclopedia Britannica says May 26th
As you noted the BBC says May 27th
The History Channel says May 26th
Admiral Tennant, then Captain Tennant, arrived at Dunkirk on May 26th according to the BBC
And, the Worcester News says that Admiral Tennant arrived on May 26th
English Heritage has an interactive timeline saying the order was given May 26th, but the evacuation started on May 27th
I cannot find anything that says Tennant arrived on May 22nd. Looks like the order was given May 26th, but the boats started moving May 27th.
Upvote:3
According to Churchill in Their Finest Hour, chapter 3, on May 22 and 23 the B.E.F. and French forces were attempting to break out of the encirclement by attacking S.W. from Arras and vicinity.
Again in chapter 4, Churchill reports that a German message was intercepted in clear at 11:42 a.m. on May 24, halting the Panzers on the line Dunkirk-Hazebrouck-Meyville.
Finally, Churchill has General Gort, on the evening of the 25th of May, ordering an abandonment of the attempt to break out of the pocket, a consolidation of the perimeter centred around Dunkirk, and a march to the sea. The motivation for this change of plan is given as the realization that the Belgian forces holding the northern flank of the perimeter were about to be overrun by German infantry.
It is certain that no evacuation would have started prior to these events having played out, there being as yet no forces on the beaches of Dunkirk to evacuate.