The Trotters pulled the plug on their fixture at just 24 hours' notice due to welfare concerns for their young squad, who have been thrust into senior action amid the off-field problems at the club.
Wanderers are awaiting the completion of a takeover by Football Ventures which had been blocked by a court order issued by Laurence Bassini, who had tried to buy the club earlier this year.
Boss Phil Parkinson named the youngest team in the club's history in their goalless draw against Coventry on August 10, and saw his side thrashed 5-0 at Tranmere on Saturday.
And a statement issued via the club's administrators said making the youngsters play again on Tuesday night "would be detrimental to both their welfare and development".
But Allardyce, who managed Wanderers between 1999 and 2007, believes their baptism of fire might help them in the future.
He said on talkSPORT: "The levels of fatigue will be quite high because they are playing at a level they have never played before.
"A crushing defeat at Tranmere would be mentally quite tough on them but when you have to grow as a player and a professional, things like this make you or break you and to stop them playing is a big blow for Bolton Wanderers, having overcome some real problems.
"It will put the focus back on them by not playing a fixture that they should have played and I think that is a real shame.
"They got a team on the pitch and they were playing and they got huge praise for the draw they got at home against Coventry, 0-0.
"When you have got the opportunity to play first-team football, which they never expected, then it would make them stronger and better players much, much quicker by this experience, even though it was a bad one.
"If they had enough players they should play the game, if they didn't have enough players then that is a different story.
"The welfare of what is my concern. There are mental strains but that is what you have to conquer as a footballer and if you are good enough you're old enough."