Not BA, but Crossair, which had a reputation for having staff who were prepared to take things less seriously (Crossair has since become Swiss).
On 1st April, on the last shuttle of the day from ZRH to GVA, just after doors closed, the PA announces:
"Welcome on board this Aeroflot flight, destination Moscow."
The passenger next to me shot up, opened the overhead locker to grab his bag and I finally put him out of his misery by reminding him of the date. Cue much laughter from the rest of the passengers on the flight.