I don't have the problem, but the symptom (both the battery and the brake warning light on) is indicative of the alternator not working/charging.
Were the battery and alternator changed because of this problem, or were they changed some time earlier?
If the problem existed before the battery and alternator were changed, then it's not likely they are the cause and there's a fault elsewhere. If the problem appeared some time after they were changed then the alternator is probably faulty. But before going too far, a question:
When you first go to start the car on one of those cold days, and turn the key to the ON position, but not start the engine, is the Battery warning light lit? This is important for the alternator to start working after the engine starts up.
Do you have a multi-meter and know how to use it? If you do, then on a morning when it's good and cold and the problem is likely to appear, open the hood, turn the key to ON, then measure the voltage at the connector on the alternator. There's two wires; one should have the full battery voltage (12+ V), and the other should have a few Volts, perhaps 1-3. If either of these isn't present, then there's a connection problem outside of the alternator.
If the two Voltages seem to be correct, and the Battery and Brake warning lights don't go out after the engine is started, then it's the alternator. In this regard, if it's an aftermarket unit or a rebuilt, problems such as this are not unusual.
In this case, whatever is causing the alternator not to work is temperature sensitive. It could be internal to the alternator, e.g. in the electronic regulator. It could also be at that connector to the alternator -- check that the connector contacts are clean and that the connector is properly seated.
