The only way you could use your Nest or any other low-voltage thermostat to control a room AC is to hook up a suitable load-rated relay inline with the unit's power supply outlet.
This is doable, but the main key to it's success or failure is if the AC unit itself resumes cooling at IT'S setpoint, after line voltage is interrupted and restored.
A lot of AC units will not even restart automatically after a "power failure" or will run cooling @ 76 degrees, or such.
For a remote thermostat to do the job, you would need to set the unit as low as it goes and allow the remote stat to cycle it on and off.
I've done this setup before and it can work!--
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