First thing is check the 24v ac. One transformer should power the air handler, the outside compressor and fan relay, and the NEST thermostat.
Remove the Nest from the baseplate and with a volt meter and power on, you should read ~24 volts on the AC scale between the Red and Blue wire or the Rh and Common. If not, your transformer is bad. A very common failure but since the NEST display still worked, 24v is more likely present.
If there is 24v's, use a wire and jump the R to the Y wire, this should turn on the compressor and compressor fan. If not then the relay is bad that resides in the outside unit.
The thermostat places the 24v's to the G wire to turn on the air handler fan. Since the Nest didn't turn off the fan I'm suspecting a bad NEST thermostat or controller board. With the NEST off the baseplate, turn on the power. If the fan still runs then I suspect that control board in the air handler is bad.
In summation what could be bad at this time with the information given, 1) 24 transformer, 2) the relay in the compressor housing, 3) the controller board, 4) the thermostat.
FYI... At your breaker panel there are two 240v breakers. One for the compressor and the other for the air handler. The breaker on the air handler is usually there when an electric heat strip is used. AC only, usually not.