20 January 1934: birth of Tom Baker, who played the Fourth Doctor from 1974 to 1981 and has reprised the role on occasion since (most recently in the BBC audio "Demon Quest" series). For me he remains the definitive Doctor. Happy birthday, Tom!
20 January 1990: death of John Maxim/John Wills, who played the Frankenstein monster in The Chase (1965) and a Cyberman in The Moonbase (1967).
ii) broadcast anniversaries
20 January 1968: broadcast of fifth episode of The Enemy of the World. Bruce has switched sides and starts to help the opposition to salamander. Meanwhile the scientists trapped underground are getting restive.
20 January 1973: broadcast of fourth episode of The Three Doctors. The Second and Third Doctors persuade Omega to return the others to Earth, and then destroy him with the Second Doctor's recorder. The Three Doctors are restored to their own timestreams.
20 January 1978: broadcast of first episode of The Armageddon Factor; first appearance of Lalla Ward (though as Princess Astra, not Romana). Atrios and Zeos are locked in war. The Doctor and Romana are captured by the Marshal but escape; Princess Astra is captured by a bloke in a cloak.
20 January 1984: broadcast of second episode of The Awakening. The Malus attempts to manifest by draining energy from the villagers and the Tardis, but the Doctor defeats it.
20 January 1996: broadcast of first episode of The Ghosts of N-Space on BBC radio. The Brigadier is visiting his Sicilian uncle whose castle is under threat from modern mobesters and ancient ghosts.
iii) dates specified in canon
20 January 1920: birth of Toshiko Sato's grandfather in Japan.
20 January 1941: Toshiko Sato and Jack meet the real Captain Jack Harkness (in the 2007 Torchwood story, Captain Jack Harkness).