ESP01 connected to Arduino can't read from UART while connected from external power source
Can I just check when you plug your 3.3V ESP into a 12V power supply it stops working?  Is there a burning smell when you do this?  Are you using anything to drop the voltage to 3.3V?

You also say you are connecting the ESP to the Arduino's 5V supply.  Normally that's not a good idea either.

Also have you tied the ground of the Arduino to the ESP/12V PSU?  If the two devices don't share a common ground then they won't agree what 0V and so won't be able to talk to each other over serial.  Just run a wire from the Gnd on the Arduino to the Gnd on the ESP and they will jump into life.