They lost to Dungannon Swifts on the away goals rule in the promotion/relegation play-off after finishing the Championship campaign second to promoted Portadown.
During his 14 years in charge at Suffolk Road, Kelly led Celtic to many successes as an intermediate club, including the 2003 Steel & Sons Cup, before taking them into the Premiership.
They spent two years in the top flight before being controversially left out of a revamped 12-team invitational set-up.
Kelly signed McAllister as a player in 2006 and also appointed him as a coach on his backroom staff.
McAllister's task will now be to rally his players for another tilt at gaining promotion.
"I feel very privileged to be offered the job and hopefully I can take the club back to the Premier League," said McAllister.
"I want to attract players to the club who will help us to realise that ambition," he added.
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