The AC should start up when the evaporator temperature sensor used by the AC is above around 4-5c. Unlikely the fan is not controlled by road speed.
If my car was stationary for considerable length of time, I can hear the fan cycle on for about a minute, and then off for next minute, then repeat, when AC is switched on with fan speed 2 (of 4). My Jazz had manual aircon, not climate control.
I recall inserting digital thermometer into centre vent many years ago to check the temperatures. It reported just under 6c when blowing cold air when set to coldest setting in spring.
Set your climate control to lowest possible temperature and verify the fan does come on and is blowing cold air out of the vents, when car is stationary. Use a thermometer to verify temperature.