Completely misses the point. The issue isn’t how Britons write the date with the month spelled out, i.e. “September 5” or “5 September.” The issue is how they abbreviate it: “9/5/2009” or “5/9/2009.” The discussion you linked to only addresses that in passing, and doesn’t cite any examples of the “American” form being used in Britain.
The spell-it-out version apparently has no standard and/or is still in transition. Did the British ever write their numerical version the way we do? When did they start writing it the other way? If so, was there a transition period and when did it end? Is it law?