Jenga: When does a player's turn begin?
It is not necessarily any player's turn, but Player A would win. Not because player B caused the tower to topple - he didn't. But according to the rules:


  Your turn ends 10 seconds after you stack your block - or as soon as the player to your left touches one.


and


  If you're the last player to stack a block without toppling the tower,
  you win!"


It doesn't say that the next player's turn begins as soon as those 10 seconds are up, but it doesn't really matter. In this case, player A successfully stacked a block without toppling the tower. It wasn't that player B caused the tower to topple, it's just that player A met the win condition.