The real Christine Keeler died four days after the second season of "The Crown" was shown on Netflix.
The comedy that the Macmillans went to see was "Beyond the Fringe", which was a huge hit on the theatre scene and which helped launch the careers of Dudley Moore and Peter Cook. It was later filmed for TV broadcasting. The other cast members are Alan Bennett and Jonathan Miller. All four also wrote the play.
When taking the photo of the Royal family, the lines said ("This earth of majesty, this seat of Mars" and "This blessed plot, this earth, this realm, this England") are from John of Gaunt's speech in Shakespeare's "The Life and Death of Richard the Second" (Act 2, Scene 1).
This episode's title is "Mystery Man". The tune played in the opening sequence is "Man of Mystery" by The Shadows.
When McMillan and his wife are having tea at 10 Downing St., the pattern of the tea service they are using is, perhaps significantly, English Renaissance by Royal Doulton.