I'm not asking when the first autoland was certified, or when one was performed in revenue airline service-- rather, when did an aircraft, any aircraft, first automatically land itself, with nobody touching any aircraft or engine control (after pushing a button a few thousand feet above runway altitude) until the aircraft had come to a stop on the runway?
Flight magazine for 9 Oct 1947 describes the C-54 flight from Stephenville, Newfoundland to Brize Norton that was automatic from the start of the takeoff roll until the touchdown-- but they left the braking to the pilot, and for all we know maybe he took the wheel too at that point, so that one doesn't qualify. But maybe they could easily have included the rollout in the automatic sequence?
(The Flight article starts on page 415, if you want to look it up in the archive at http://www.flightglobal.com )
The article also says "a Lancaster at the Blind Landing Experimental Unit, Martlesham, has already executed some 200 automatic approaches, many of which have embraced automatic landing." But it doesn't say automatic rollout, and one suspects it wasn't.