My personal thought is:
November 11, 1918
My logic:
I've always stuck with November 11th as I like the fact that it's Armitice/Veteran's Day.
Now the year. In "Moonraker", Capter 1 - Secret Paper-Work, we learn a bit more about Bond personally. At one point he muses over being taken off active duty in the Double-O section and the mandatory age of 45 and that being eight years away.
Moonraker is set in the late summer of 1955. So, if in eight years time from then (1963) Bond would be turning 45, the simple math points to 1918 and Bond is presently 36 at the time of Moonraker. He hasn't had his birthday yet.
Yes, Fleming transposed and often changed dates around as he really didn't care or considered Bond ageless, but that's just the one bit from Fleming that actually gives some solid numbers.
Then again, Hey, it's Bond...Nobody said there'd be math.