OldSteve:
That Schottky diode actually appears to be between USB +5V and the regulator's output, not Vin. Current can only flow from +5V to Vin as I described earlier. Vin only connects to the regulator's input. (I think you misread the line from the Schottky anode as joining Vin, but it doesn't. There's no blob where they cross. And that would cause a whole raft of other problems if they were joined at that point.)
Oops. You're right. I didn't look carefully enough.
But if the regulator has failed, but not shorted, the Nano should still work if powered by the USB. If the Schottky diode has failed the Nano should still work if powered via Vin. (If I read the thing correctly now, that is).