The Royal Variety Show" and second, it didn't take place, like the poster says, at the London Palladium, but rather at the Prince of Wales Theatre, November 4th, 1963 (There more inaccuracies concerning the poster, take a look at the comments section of this post). Many people think that John Lennon's famous "rattle yer jewellery" remark was aimed at Queen Elizabeth, but guess what? She wasn't attending the show. The Queen Mother and Princess Margaret was there, though. In the Rutles retelling of the story, it's an impersonator of Queen Elizabeth who is attending, or "Barry's mum", if you wish.
Apart from the aforementioned poster, in 1963 a 32-page 7" x 9" softback souvenir photo book named "The Beatles By Royal Command" was produced in the UK by the Daily Mirror newspaper, documenting their Royal Variety Show performance from arrival through rehearsals to the aftershow with a great live shot colour centrefold.
The Beatles did play a gig at the London Palladium in 1963. That happened on October 13th, but that was for the TV show "Val Parnell's Sunday Night at the London Palladium". Very few episodes of this programme have survived (5 out of original run's 126), the Beatles' episode being one of the lost ones (contrary to what Wikipedia says).