It is a highly unlikely scenario where the PU will not see the contact AND there is an extended range of space where the contact COULD have happened.

From the PU's perspective (no contact with the balls seen in each case):

The batter swings at the pitch, hits the ball down, and the ball scoots out into the infield (or out into foul territory). BU kills the ball. Contact was in the box, foul ball.

The batter hits the ball out in front of the plate (or a dribbler moving out into the infield), the batter-runner begins running, BU kills the ball. Since the ball was out of the batter's box, contact was out of the batter's box, batter out.

The PU is not blind and is not unaware of where the batter-runner is and where the ball is; he just did not see the actual contact. 999 times out of 1000, though, he will know that IF there was contact, WHERE the BR was when it happened. And, he'll know this with more certainty than the BU will if the contact is in the vicinity of the batter's box.

Jasper- The other thing that needs to be mentioned here is that this is not an "appeal" play. An appeal is a specific play that is defined by rule. This play does not meet that definition.

What we have here is the defense telling the PU that they saw something s/he may have missed (the B/R making contact with the ball) and asking that the PU confer with his partner to see if the BU saw the contact. Since the BU saw the contact, the play reverts back to the point of the contact. There is no "timing" involved.

The rest of the replies have covered how the play should have been handled by the BU when s/he saw the runner contact the ball.