Cooling fan does not turn on. should it run all the time or only after it heats up?

The fan will run on for approx 1 minute after it has been turned off. This is to cool the unit down and clear any fumes or gas.

Of course not. You may have a defective blower relay.

I'm not sure exactly how this applies specifically in a car, but on a home heating and cooling (air conditioning) thermostat, it is often associated with the fan or blower setting.

If for a system fan:

"Auto On" setting in a heating / cooling system would start the fan when the temperature climbed above or dropped below the setting you on the thermostat and start the heat or cooling source. Once the thermometer indicates the desired temperature had been reached, the heating / cooling source would be stopped and the fan would turn off.

"On" would run the fan all the time, regardless if the thermostat tired to start the heat or cooling source. This would result in constant circulation of the air in the space(s) and tend to keep warm or cool spots (rooms) from being created.

Seems like the fan relay has stuck (Likely to be on the main indoor PC board) or if the machine was being used on heating mode, the indoor pipe sensor may have failed causing the fan to continue to run. Most air cons on heating mode, will keep the fan running for a few minutes after you switch off on heating mode to cool down the indoor unit before it turns off completely. The pipe sensor signals the main PC board to tell it that the unit has cooled down and can now turn off completely. If either of the above faults occur, the indoor fan will keep running all the time.

Is your radiator cooling fan operating when the A/C is turned on? The cooling fan should be running at all times with the A/C on. If it does not, then your problem is being caused by a bad cooling fan motor, a bad cooling fan relay, a broken fan motor circuit, or broken fan control circuit.

the cooling fan running for a short time after turning off is normal.as far as the NO HEAT is concerned have the topmost hot hose clip undone and vent any airlock you may have.(pref a hose that is attached to the heater inside the engine bay).

the coolings fans are designed to run for a little while to dispense heat they dont do it all the time turn usually during higher hot days w/ac on or higher hot days goes off automatically

Circ ==== Circulate.. To move "Air" around.

Some cars will do that for a limited time. The reason is the heat builds up and using a term "heat sink", the car is now sitting and there is no air flowing to cool the parts. The water is not pumping to cool the engine, there is no air flow past the exhaust and the exhaust heat is trapped.
The cooling system is marginal so the automakers let the cooling fan run until the motor cools down. There is an extreme, I would think over 15 minutes. Call a Cooper dealer to check how long this fan should run.

On some cars the fans will run anytime the heating or cooling system is in the defrost mode try turning it to the vent mode or floor and see if one or both go off