Allen confirmed the cancellations during an appearance at a Broadcasting Press Guild lunch, according to Broadcast journalist Jake Kanter.
Elton's The Wright Way was savaged by critics when it launched earlier this year.
The Guardian said that it would have looked "badly dated in the 1970s", while The Independent billed the comedy as "occasionally so groan-inducing that you want to gather a mob with torches and pitchforks".
Perkins's Heading Out received a warmer critical response, but struggled in the ratings, falling beneath BBC Two's slot average at 10pm.
The BBC comedy team has commissioned a second series of new sitcom Count Arthur Strong and has also green-lit a new comedy from Getting On duo Jo Scanlan and Vicki Pepperdine.
Watch a clip of The Wright Way: