Had a similar problem, only had hot air when travelling, if I stopped or turned the fan speed up passed No1, the air went cold very quickly.
Sorted it by replacing the auxilliary water pump (under the expansion tank).
As I had a spare I just replaced it. But on inspection it was only full of crud, which stopped the impeller turning. Impeller is not connected to the motor shaft, it runs via a magnet which is rotating in reaction to the motor rotor. (or something like that)
It can be serviced, undo the 4 cross head screws, pull off the the pump cover with the inlet/outlet pipes connect to. (orange seal situated under)
spin the impeller, if it turns easily its not the problem, if it's rough or jammed, just pull it out.
Clean the crud out of the bowl and off the magnet and re-assemble