Radiator fan not spinning

Hello and welcome to 2CarPros. In the case of your Mini the fan only operates when the computer turns it on. Either in response to a cooling request because the coolant is hot enough to trigger it or if you turn on the AC and it needs to move air across the condenser.
During normal driving the fan will normally stay off as there is enough air movement to keep things cool. It has a two speed fan that turns on at low speed at 105 C coolant temperature and turns off when the temperature drops to or below 101 C. It will switch on the low speed and then switch to high speed at 112 C and once the temperature drops by about 4 C the fan drops to low speed.
From your description and because it isn't overheating I suspect you don't have a problem.

However, if you wish to test it, try turning on the AC and let it run a bit. The fan will start on low speed. Another test would be to disconnect the coolant temperature sensor and put a jumper across the black wire with white stripe and the yellow wire with red stripe on the plug you removed. That makes the computer turn on the fan to be sure the engine doesn't overheat. However this test will very likely set a code and turn on the check engine light.

Thank you so much for your reply! I've been freaking out about it and you've definitely put my mind at ease! Legend.

Another quick question, is there a water tank in the engine bay that needs refilling regularly? (Other than the windscreen washer water)
Or does the engine just use coolant, as I can see a tank for that?
Thanks

Depending on which body the car has it can have either a pressure bottle or a recovery tank. Some even had both just to make it interesting. The pressure bottles use a screw on cap. The recovery bottle is a bit different, it just catches excess coolant that might get pushed out, then it draws it back in when the engine cools off.
They will have a mark on the side that shows the level they should be at with a cold engine. In theory you should never need to add coolant to the pressure bottle system as it is sealed. However after many years there might be some tiny leaks and you may need to add a bit because of them. It shouldn't be a constant thing though. Maybe a cup in a year or two.
The recovery bottle is again a bit different. It isn't a sealed system so some water can evaporate out of it, again it shouldn't be much needed. If you add the proper coolant to top it off and then need to add mare after a week then you might have an issue. An easy way to check this is to add the coolant to the mark cold. Write it down, I like to keep a couple of those mini notebooks in the glove box and write down when I check the oil and coolant levels, tire pressures and such, I'll note if I have to add anything like oil/coolant and how much. That way if something changes I can catch it. There are phone apps that do this as well but for me the book is faster.