The Hinckley United stopper has agreed to become Greene's first recruit of the summer following the Knitters' relegation from Blue Square Bet North.
The former Manchester City trainee, who has since played for a number of non-league clubs, put pen-to-paper on Saturday morning under the watchful eye of club secretary John Blackwell.
Timms (27) has endured off-field problems in the past but Greene is confident that he has recruited a reformed character and one of non-league's finest goalkeeping prospects.
Greene told bufc.co.uk: "Ashley is a top goalkeeper and we are fortunate to have him because other clubs were seriously interested in his services.
"I am only interested in what he does on the pitch and I am confident he will deliver the goods."
Greene has already confirmed that Sam Vince will act as the Pilgrims' goalkeeping understudy in 2013-2014.