Why does the car heating system won't start for the first 15-20 minutes of driving in the winter?

Most cars are heated using heat from the engine, usually with a smaller radiator used to transfer heat from the cooling system to the internal heating system. The colder it is outside the longer it takes the engine to get up to temperature.

That said, 15-20 minutes is far too long and I'd check the coolant level in your car as it may be low, meaning the engine could be too hot but because the coolant isn't circulating properly it's not reaching the matrix radiator that's supposed to warm your air.