The new schematic makes a little sense but it still won't work.
The top transistor (Q5?) is wired wrongly to be a 5V regulator. It should be wired with the incoming voltage to the collector, +5V taken from the emitter, have the Zener diode from base to GND and the bias resistor from collector to base. Just about everything is wrong!
The LED circuit will light when the battery is on charge although most of the charging current will flow through the base-emitter junctions which is not a good idea. I'm not sure what D1, D2, D3 and D4 are there for, they seem to have no purpose. If they are to protect charging circuits when the power is off but a charged battery is still fitted, the 1.2V from the battery is low enough not to do damage anyway. There is no charge current regulation so either this is a failed attempt at designing a charger or the schematic copied from a poorly designed one.
Given the 5V to start with and the maximum current limited to about 3.2mA, it would make more sense to simply wire the LED in series with the battery.
Brian.