Your device needs 12 volts, and it will get it from the battery, which is being charged by the charger. When the AC cuts off, the charger cuts off and the battery alone powers the device.

But it’s the details….

A lead acid battery will develop 14 volts (13.5 to 14.5) when being charged. Will your device operate ok at that voltage?

Does the charger taper the charge down to a trickle charge when the battery is fully charged? you need to check this, otherwise you could have battery damage.

When AC power cuts off, the battery will drop to about 12 volts (perhaps 12.5v) , gradually decreasing as the battery becomes discharged. At some point, it will be down to 10 volts, when it is near the end of its capacity. The question is how low can your device operate? 11v? 10v? Will the device be damaged by operating at a reduced voltage? The exact curve the battery follows as it discharges depends on exactly how the battery was built, ie, it will vary from type to type.

When the AC power is restored, the voltage on the battery will go back to 14 volts, how quickly is determined by how much it was discharged.

If you discharge the battery all the way to 0, you could damage the battery.

edit: be careful with lead acid batteries indoors, they can emit toxic fumes. Use only the sealed type, and avoid overcharging. Personally, I’d stay away from them totally and use some other type of rechargeable battery.