The base of the PNP sits one diode drop below the emitter, so it's going to turn on right away, only the 10k base resistor severely limits its base current so in reality the LED will probably never light up. In reality it will probably teeter on the threshold/toe. The battery voltage is irrelevant. This won't work as either a charger or battery tester...
100mV of difference between the junctions is enough to pull 10uA through the base, which is going to give 1-2mA to the LED. Very easily achievable.
But the circuit is really just an indicator that current is flowing, not a charging circuit.
The reason that I'm using 5V regulator is because power supply isn't very stable, and it's voltage is very high while not connected, and low when connected to discharged battery.
The problem isn't that you're using a reference voltage, it's that LM7805 is a terrible choice of device for this purpose. It's a regulator, not a reference.
Your circuit won't start because your reference voltage is referenced to the circuit ground, while you are trying to compare it to a voltage that is referenced to the positive supply rail (ignoring the diode). When your circuit turns off, the voltage is going to shoot up because you are going to unload the supply rail AND because as soon as you disconnect it, the bias current of the comparator is going to push the potential upwards and reverse bias the diode.
On top of everything, you're relying on the supply to limit the current that is going to be shoved into the battery. Bad idea.
I realize that what I just wrote may be a bit overwhelming for a novice to fully grasp, so I'm going to simplify it: The circuit is terrible in every way, I suggest you find a new one.