Rowan Atkinson had actually been a serious contender for role of The Doctor in the regular TV series.
Despite it being satire and thus non-canon, Joanna Lumley was the only woman to play The Doctor on-screen before the casting of Jodie Whittaker in 2017. Coincidentally, in both cases they are considered The 13th Doctor.
The appearance of Joanna Lumley is a reference to an infamous mid-80s rumor that the Doctor would be played by Joanna Lumley.
The TARDIS console seen in the film is not actually the BBC prop (which had been destroyed) but was instead constructed for use in amateur fan videos based on Doctor Who.