This suggests to me a story in which Carry On Doctor Who is being filmed and the aliens ARE invading under cover of the film (the old 'the monsters are real not actors dressed as monsters' schtick), and the world is heroically saved by Sid James. Interestingly, there was apparently going to be a Carry On Spacemen which never happened in 1961 http://en.wikipedia.org/wiki/Carry_On_Spaceman
I had hoped that the world could be heroically saved by Sid James, Mandy Rice-Davies and John Profumo with Stephen Ward in the pay of the aliens - Profumo having to take the disgrace but knowing that he saved the Earth. However, although the Profumo Affair came to a head in 1963 it was before Dr Who started, so it would mean moving the start of Doctor Who back 2 or 3 years (on the basis that it is more acceptable to move the start of a TV series than the fall of a government).
Strangely, I prefer Sid James for the Doctor, over Kenneth Williams. He could regenerate into Jim Dale for the sequel. Barbara Windsor as Leela and Hattie Jacques as Romana. Williams as the head of UNIT perhaps.