Check the main thermostat, simply let it run briefly (5 minutes) after the cold start and then check the lower inlet to the radiator to see whether warm water is already flowing to the radiator. If everything is OK, you can "disconnect" the hose on the transmission cooler and see if the temperatures are higher during a test drive. The transmission oil cooler is only available complete, I haven't seen the thermostats individually yet.
Edit: Link entfernt und User ebenfalls.