Widow Twankey was a character in the pantomime Aladdin.
During the course of other adventures, the Seventh Doctor remembered Mike Yates playing Widow Twankey in the UNIT staff Christmas panto of Aladdin. Jo Grant played Aladdin, while the Third Doctor himself played the villain. (PROSE: Timewyrm: Revelation, No Future)
In the Brigadier's great-uncle Mario's house in Italy, Sarah Jane Smith found a portrait that appeared to be the Brigadier himself in drag, who she felt resembled Widow Twankey. The figure looked like a "severe matron [...] with hair parted in the middle and sporting utterly inappropriate ringlets". She giggled for an hour after seeing it. (PROSE: The Ghosts of N-Space [+]Loading...["The Ghosts of N-Space (novelisation)"])
Behind the scenes
Presumably, this portrait actually depicts the Brigadier's Granny MacDougal, who is Mario's cousin.
In the pantomime, Widow Twankey is played by an older man. Peter Butterworth, Ian McKellen, Christopher Biggins, Barry Howard, Eric Potts and Clive Rowe have all played her.