In The War Machines, both WOTAN and Professor Brett, under WOTAN's influence, refer to the Doctor as Dr Who. This is the only time he is referred to as this, although he adopts the German alias Dr. Von Wer in The Highlanders and signs himself as 'Dr W.' in The Underwater Menace.
One possible explanation for WOTAN's error is that it has access to a report written by Group Captain Gilmore in 1963 (Remembrance of the Daleks), in which the Doctor's identity is unknown. Misinterpreting a phrase in the report, WOTAN believes 'Who' to be the Doctor's surname (Or WOTAN could have bugged the phone lines of Ian, Barabara, Ben, Polly or Gilmore, which would give the name of the TARDIS and the pet name 'Dr Who'.)
Alternatively, WOTAN might file all unknown surnames as 'Who'. At some stage the Doctor becomes aware of this, and, finding it amusing, uses the false name on several occasions thereafter. Alternatively, the Doctor's Gallifreyan name is long and almost unpronounceable (cf. Romana), but it begins with the syllable 'Who'.