Jimmy Gilligan, who will revert back to assistant manager, made a brave decision in his final team selection before handing the reins to Wilson, dropping top scorer Izale McLeod for Saturday's FA Cup match-winner Allan Smart.
But it was Peterborough who carved out the first chance of the match for an unmarked Callum Willock in the sixth minute, glancing his header wide from skipper Curtis Woodhouse's teasing cross.
The Dons rarely threatened in a lacklustre first half an hour and they were made to pay for their sluggishness when Constantine opened his account for Posh with a long-range thunderbolt in the 25th minute.
The ball fell nicely for Constantine 25-yards out and he broke the deadlock seemingly out of nothing with a sweet first-time strike into the top corner.
Malvin Kamara threatened to equalise for the home side moments before the interval as the Dons woke up, but after working a good opening for himself his weak effort was cleared off the line by Sean St. Ledger.
And then seconds later the in-form Small was put clean through, but his powerful low drive was saved by the legs of Mark Tyler.
Milton Keynes started the second half as they had done the first but midfield wizard Gary Smith dragged his team forward and conjured up an equaliser out of nowhere in the 52nd minute.
The former Middlesbrough man picked up the ball on the halfway line and showed enormous strength, skill and determination to burst into the box and cross for the grateful Small to convert from four yards.
Smith's enthusiasm seemed to rub off on his team-mates thereafter as the Dons began to play some neat football and put some real pressure on the Posh rearguard for the first time in the match.
It didn't last for long though as a disappointing Smart was hauled off for nine-goal McLeod midway through the second half.
And neither he or any of the players on the pitch could carve out a real chance in a dull end to the match, showing why both teams are struggling near the bottom of the table, and Wilson how much work he has to do.