I have a E46 vert and this caused issues with the auto aircon temp until it went back up to the classic 40 degrees. The air con uses the outside and inside temperature I believe. All sorted once sensor was replaced.
On a hot day, if you've set the temperature to a cooler temperature than the temperature inside the car, the car's computer checks the temperature outside the car. If the temperature outside the car is cooler, the car will try to cool the car by simply blowing in outside air and will not turn on the air conditioner. It's trying to save you energy.
If your outside "ambient temperature sensor" has been ripped off, and is missing, the resistance between the two wires that lead to it will be infinite (the circuit is not complete). The car interprets this to be very cold, and will display a very cold temperature (-40 degrees) on the display inside the car. The car thinks that because the resistance is high, the outside air is cold, so it will just blow outside air into the car, and will not turn on the air conditioner. The air could be very hot, which it will be on a hot day. The car doesn't know this.