I'm powering the Arduino through its external power supply.
If I connect the positive and negative lines of my power adapter to one battery, I get 12V, and can run the Arduino, but the voltage I read from the pot on the UNO is incorrect.
If I connect the positive and negative lines of my power adapter to different batteries, then I have 24V at the power adapter.
I did not run a ground to the UNO. Should have I?
English words are not the best way to explain exactly how you have stuff wired up. A schematic drawing is the universal language of electronic circuits and wiring and should be used if at all possible when seeking accurate help with problems. So even if you have to hand draw it and take a picture and post that, it's usually much better then hoping others understand what you done with your wiring.