Overthrows with Pressure

[KJUmp]

I'm with Tuss &MrUmp in what they say and their reasoning. I was taught the same way, and have handled that sitch in that manner for a lot of years.
Whether or not BU wants to make that call in fair or foul is really a matter of comfort/personal preference...there's no right or wrong there.
For me it really depends on the angle F4 is taking as he moves to his left to field the ball and make the throw. But let's look at the second part of the play...the overthrow.
BU's first responsibility is the BR...and anything and everything that can happen concerning him. Did he make an attempt to go to 2nd, does he decide to go to 2nd, does he start to go to 2nd and then decides to go back to 1st, possible obstruction? There's no reason for BU to concern himself with the overthrow. One of two things are going to hapen to the ball...(1) it becomes dead or (2) it stays live and in play. PU (who would be moving up the line in the OP's sitch) can easily handle both 1&2 as he's got the play in front of him and can easily move to any position necessary to rule on the ball becoming dead.
BU needs to read & react to the overthrow. If BR goes to 2nd cut behind him and onto the infield grass (if starting from fair territory) pick-up the ball over his right shoulder and let the thorw dictate his angle to make the call at 2nd. If he started in foul territory...it's a straight sprint to the infield grass...and then follows the same process. This should not be a difficult move for BU 99.999% of the time if he has moderate to decent "wheels". If the ball stays live, and BR stated to 2nd then goes back to 1st BR is in position to make the call at 1st should there be a play.