Aircondioniong not working, no fan or cold air

Low coolant, if there is an air pocket in the engine it will make it to the heater core. Once there , the air coming out is cold/cool. Once the coolant fills the heater core, it gets hot. If the coolant is fine hot or cold, then have the HVAC system scanned.

warmth comes from the heater in the car
if it is climate control check the climate control module
check for heater control in the vent settings as the hot water tap may not be turning on

most likely the hot/cold blend door actuator is not working and these are all electric not vacuum controlled

If you have an automatic climate system where you can set temp the the compressor will run at some of the middle settings. The system blends cold and warm air to obtain the desired temp setting. If it does not have automatic control you have another issue.

Could be thermostat needs to be replaced.

Is the compressor cutting out? It could be the compressor clutch relay is going bad.

You may need to reset basic settings for the control module using a scan tool , reading the logged faults stored in the control module is needeed also .

After warmup it still blows cold, or only when cold? Normally the fan will operate independently of engine temp except in some climate controlled vehicles. As long as it blows warm air after engine is at temp that's a good thing. the heater uses hot water from the engine. Obviously the heater can't be warm before the engine is. If it's cold after the engine is hot, there are vacuum controlled "motors" on top of the heater box under the dash that move doors that control heat. Either one of the control motors or one of the doors have stuck or failed. Many also have a control valve on one of the heater hoses under the hood as well. If the engine comes up to temp then the thermostat is good (should be at about 210f) Only remaining item is the core itself which may be internally or externally obstructed.
All this being said, if it's good when the engine is up to temp, leave it alone...it's fine.

This problem is caused by what is called the air temp control blend door, either the door is sticking in the duct or the electric actuator that moves it is defective, the actuator is the common part that fails, the bad news is the dash assembly must be removed to replace the parts.

Replace the temp control on this car, it should be electrical control not cables.