Alex could have arranged for the game to go to the school after he returned so it was still there for the kids. He would have known who the kids were and how they came across the game. Their lives wouldn't have turned out any different after Alex's return so there's no reason they wouldn't have still ended up getting detention and playing the game.
However, Alex could have just carried on his life and done anything with the game. After completing the game, time is reset to the point before the game was started with the players retaining their memories of the game (provided they're even alive at the time the game started).
In the first movie, Alan and Sarah remember everything because they started the game and Judy and Peter aren't even born yet.
Grown up Alan and Sarah meet up with Judy, Peter, and their parents before the parents were killed and before the kids join the game thus changing Judy and Peter's future from then on.
They won't remember Alan, Sarah or their experience with Jumanji at this point because it hasn't happened for them yet (and never will).
There's no reason to suspect time travel works differently in each movie though.
One significant difference, and I beleive it to be the main reason, is that in the second movie, all players were inside the game and were sent back out from inside the game.
That probably has a lot to do with them remembering everything when they returned and not being affected by the changes in the timeline. They had become part of the game rather than just playing it.
In the case of Judy and Peter, they were just rewritten when the timeline reset.
I think if they had ended up inside the game, it's quite possible they would have come back out to the time they went in and found themselves in the changed timeline with their parents still alive etc. and with their memories from their original timeline and their time with Jumanji.