Hi Damjan,

I am afraid this will not work.
First of all, a 4.8V supply in case of power down is a bit low for your circuits.
Second, the Ni-Mh cell is connected, in normal operation, to 5,35V. (0,1 Amps load) I think that to much.

It's better to put a diode between the 7806 and the circuit. Your circuit then has 6-0,7 = 5,3 V. Use a 6V battery, and a diode between the battery and the circuit. Again 5.3 volts on your circuits. To keep the battery charched put a diode between the 7806 input (the not stable side) and the battery with a diode and a resistor in series. The resistor must limit the current to 1 milli Amps.

Mail me if you need the scematics.

Regards
Dick