Based on the way you describe the play...
If the ball is judged to have hit the hands of the batter, the umpire will immediately call, "Time!" If a check swing and the plate umpire has a swing, he will signal and call the strike and since the ball is dead, the batter is out. Should the plate umpire have no swing, then he will immediately go to the appropriate wing (First or Third base) umpire for an appeal. This is the one time he will not wait for the defense to ask for the swing appeal. If the appeal is "Yes, he did.", then it is strike three, batter out as above. If it is "No, he didn't.", then the batter is awarded first base.
The hands are never an "extension" of the bat. Any pitch that touches a batter is a dead ball without exception. Now if the pitch is in the strike zone, whether the the pitch touches the batter is moot; the pitch should be ruled a strike.
Hands part of bat is as much as incorrect rules undertstanding as balk is a pitcher deceivng the runner r tie goes to the runner. Sandlot rules that just are not true. Nor correct.