Hi there,
I have a nano from which I am trying to control a PWM mosfet module.
When I checked the output of the nano prior to connecting my circuit, these were the results:
9V to Vin --> 4.01V from 5V pin
12V to Vin --> 4.05V from 5V pin
USB power --> 4.68V from 5V pin
I tried another nano and it works fine (5.07V out), as does an uno (5.001V out).
Is the voltage regulator on this nano toast, or is it a problem I can fix in some other way?
The USB power resulting into 4.68V on the 5V pin is normal. There is a diode between the USB 5V and the 5V pin.
When applying 9V or 12V, the voltage regulator is used. It is only a small voltage regulator. Maybe something is connected to the Nano board that needs too much current. Maybe the voltage regulator is toast after getting too hot, maybe it is a 4V voltage regulator 
maybe you have a very cheap clone and that is not even a voltage regulator 
Remove everything from the Nano board. Apply 9V (not from a weak battery) and check the 5V pin. If it is near 5V, then everything is okay. If not, then get rid of the board, because you don't know what else is fried.
The voltages were measured without any load on the board whatsoever (nothing connected - just Vin/Gnd and measuring voltage from 5V/Gnd with multimeter).
It is a knockoff board, so it having a 4V regulator definitely seems plausible to me.
I haven't had any other issues with the board and have been using it exclusively for all of my test projects and arduino practice.
I was just making fun of your knockoff board.
If you bought the cheapest Nano board, then the manufacturer might have taking just any voltage regulator that they have lying around. If it works a little then it is sold. You would be the first with a 4V regulator. It is not completely impossible.
It is more likely that the voltage regulator got too hot or that you applied a negative voltage to VIN.