From the rules, the hand is over.
When you have one card left, you must yell "UNO" (meaning one).
Failure to do this results in you having to pick two cards from the
DRAW pile. That is, of course if you get caught by the other players.
Once a player has no cards left, the hand is over. Points are scored
(see scoring section) and you start over again. That's UNO in a
Further, you only have until the next player starts playing a card before it is to late to call them out. In this scenario once Player 2 started his 3rd turn (by starting the motion to play his last card), it was too late to call him out.
A player who forgets to say UNO before his card touches the discard
pile, but "catches" himself before any other player catches him, is
safe and is not subject to the penalty. You may not catch a player for
failure to say UNO until his second to last card touches the DISCARD
pile. Also, you may not catch a player for failure to say UNO after
the next player begins his turn. "Beginning a turn is defines as
either drawing a card from the DRAW pile or drawing a card from your
hand to play. If the last card played in a hand is a Draw Two or Wild
Draw Four card, the next player must draw the two or four cards. These
cards are counted when points are totaled. If no one is out of cards
by the time the DRAW pile is depleted, reshuffle and continue play.
In order to call out Player2 and force him to draw cards for not saying 'UNO', Player 1 would have had to say it directly after his second to last card was played (touched the discard pile) but before he started to play his last card. Incredibly short, but possible.