it depends on what year the car was made. is the fan belt used to drive a fan by the radiator, or is it an electric fan?
my mark 1 ran OK without the fan blades fitted, which I tried when a friend said it helped the car warm up in the winter. I put the fan blades back on when a hose burst as I was blasting down the motorway LOL
if its electric, there is a thermostatic switch with 2 wires to it, probably on top of the thermostat housing. they don't often need to cut in in UK weather, so the switch or the fan could have failed.
you can test the fan by shorting the 2 wires together (if you are confident you know what you are doing of course!). some work without the ignition on, some need it switched on.