The War room will have an official, usually a retired one, watching each game. So if 4 games are played at once, 4 officials will be seated at individual monitors watching one game each. Since they will be retired or even active officials, they will know what to be looking for during the live play. If they see something in that live play, they can signal the on field official that they are going to review it, that way they can stop the play before the next play starts.

What they will be reviewing is the same as it has been, it's just that instead of the referee on the field reviewing the play over at a monitor on the sideline, an official or officials in Greensboro will be making the call, and that will speed up the game.

As for the scenario you described, that will always be that the play would count since the play started before the review was called for.