3 April 1924: birth of Peter Hawkins, the original voice of the Daleks and Cybermen.
3 April 1929: birth of Michael Hayes, who directed The Androids of Tara (Fourth Doctor, 1978), The Armageddon Factor (Fourth Doctor, 1979), and City of Death (Fourth Doctor, 1979).
3 April 2008: death of Johnny Byrne, who wrote The Keeper of Traken (Fourth Doctor, 1981), Arc of Infinity (Fifth Doctor, 1983), and Warriors of the Deep (Fifth Doctor, 1984).
ii) broadcast anniversaries
3 April 1965: broadcast of "The Knight of Jaffa", second episode of the story we now call The Crusade. King Richard asks the Doctor and Vicki to join his court, and sends Ian to rescue Barbara, who meantime has been handed over to the evil El Akir.
3 April 1971: broadcast of fourth episode of The Claws of Axos. The Doctor sends Axos - and the Master? - into a perpetual timeloop, but is himself unable to leave Earth.
3 April 1977: broadcast of Whose Doctor Who, a BBC documentary about the show, presented by Melvyn Bragg.
3 April 2010: broadcast of The Eleventh Hour, first story in Season Five of New Who; first full episode for Matt Smith as the Eleventh Doctor, first appearances of Karen Gillan as Amy Pond and Arthur Darvill as Rory Williams.
iii) dates specified in-universe
3 April 1926: date on the Illustrated London News found by the Third Doctor and Jo Grant on the S.S. Bernice, though in fact the crew and passengers think it is June, little realising that in fact they are on display in Vorg’s MiniScope in Carnival of Monsters (1973).
3 April 1938: date that Rory Williams is transported back to in The Angels Take Manhattan (Eleventh Doctor, 2012)