You can connect a battery to the DC side of the NAT router directly and have that be its primary power supply. You would discontinue use of the router's own power block, and use an appropriate off-the-shelf battery charger for that battery type. This battery charger will be perfectly safe if UL listed, and will simply plug into the wall.
Secure the lid with the M3 screws, Working: In normal operations, the battery will charge through the balancer circuit by the wall powered adaptor and adaptor will power the router. When there is no power the UPS comes into the picture and runs the router without any delay. It's a safe and clean way of recycling your old laptop battery.
The goal is to run the router on DC power straight from the 12v battery avoiding the DC -> AC -> DC conversion that would take place using an inverter in between and the losses that come with each conversion. These losses contribute to shorter run times.
If this loses power, the Internet will be inaccessible. Fortunately most such devices also take 12VDC power (check this), and likewise can be run off the same battery. The market is thick with routers and modems that run on 12VDC; it's easy to buy ones that do.
You would discontinue use of the router's own power block, and use an appropriate off-the-shelf battery charger for that battery type. This battery charger will be perfectly safe if UL listed, and will simply plug into the wall. The AC side will be protected and you'll have access to the safe low voltage side only.
Use required protective gears for your safety. Start from the one side of the battery with a flat tool such as a scalpel or a flat screwdriver and try to lift it from one side. Uniformly apply force by slowly moving ahead with your screwdriver tool and break open the plastic case.