Remember, if the fan resistor has failed, the high speed setting for the fan will still work; the high speed fan will still come on at a higher temperature and stop the engine overheating. What year is your car? There are two different fan wiring circuits, one for cars up to mid 2003, which had the control relay in the fan assembly, and the later one with the control relay in the fuse box.
Check your cooling fan fuse (it's F5 in the under-bonnet fuse box on my car (a 2004 Cooper), but it might have been different on the earlier MINIs). Does your car have a power steering cooling fan? (only on the Cooper S in the UK?)
Early cars shared the circuit for the PS fan and radiator fan. Any garbage sucked up by the PS fan and jamming it will cause the fuse to blow and stop the radiator fan working. This is not a problem with post 2003 cars.
Try testing the fan assembly. The plug connecting the fan wiring is on the passenger side for your car, mounted to a bracket that also supports a refrigerant connection if you have air conditioning. It's near the bonnet catch. The plug has four pins but one is unused. On my car, the three wires are a brown ground wire, a red/blue wire for the full speed circuit and a red/green wire for the low speed circuit. Disconnect the plug and test the resistance between the fan pins and the ground connection with a multimeter. If both the ground to low speed pin and the ground to high speed pin are open circuit then this is consistent with an open circuit fan motor or a break in the wiring between the resistor and the motor. If only the ground to low speed pin is open circuit, then this is consistent with a failed resistor or a break in the wiring to the resistor.