12v battery and charger on arduino uno?
Yes you can probably do that and basically that's how most (non professional) UPSs work as well. They consist of a Lead-Acid battery, a charger which charges that battery continuously, an inverter (to make mains voltage from 12V) and some control circuit to switch to the power from the inverter when the mains is off line.

If you're using a Lead-Acid battery then it is OK to charge it continuously. The charging will stop or switch to a very small charging current when the battery is full. As a Lead-Acid battery charges its voltage will rise and at about 13.8 - 14 V it is full. As most chargers supply not more than 14 V charging will stop by itself.

If you do not have a charger yet I suggest to select one with a low charging current as that will save power. Initial charging (when the battery is empty) will take long but as the system will mainly be mains powered, that isn't an issue.

Then indeed just supply the Arduino directly from the battery and you're done.