Does he win automatically, before Player 1 goes into his Standby Phase, or does it have to wait, until the Draw/Standby phase of Player 2?
Exodia's win condition (note the word, it's not an effect), triggers immediately and not only in your turn, so you would win during the start of the Draw Phase of player's 1 turn.
Do note that this is not so common to happen, but it could eventually.
From Exodia's rulings, emphasis mine:
At the start of the Duel, if the player going second starts off with all five pieces of "Exodia the Forbidden One" in his hand, then he wins the Duel before the first player draws his card. The same applies if the player going first starts off with all five pieces. If both players have all five pieces of "Exodia the Forbidden One" in their hands at the start of the Duel, then the Duel ends in a DRAW.