This actually happened can't make this up. Just slowing down and now have some time to read the board and post this.
I don't think the level of the game makes a difference on the ruling, probably the same in all rule sets.
12yr All-Star game. 5th inning 2 out Runner at 3rd score tied 4-4.
Batter swings, ball bounces off the plate and up like a short fly ball.
F1 catches it on a short hop.
F1 runs

toward home in an attempt to tag R3 who is trying to score.
R3 does a sit down slide and F1 misses the tag on R3's back by about six inches, SAFE 5-4 visitors.
I check the field all action has stopped so I call time and proceed to cleanoff the plate.
Coaches, players and fans say nothing, no complaints on any thing. My partner, who is a 15 year old young man, says nothing.
Batter steps in the box and hits the first pitch to F5.
Throw to F3 out inning over.
I pick up the bat and hand it the the 3B coach and he tells me the batter that just batted should have been on first, he is the one that hit the ball off plate to the pitcher and didn't run.

Too late now I tell him, play stands run counts. I said nothing. I never even thought

about the batter still being at the plate for a 2nd at bat.
When should have my partner called the batter out for abandoning 1B or does he say nothing and wait for the defense to appeal or ask?
I say when batter stepped in to the box the 2nd time out for abandoning his base.