Protecting a wireless router from voltage fluctuation
A simple large capacitor placed in parallel with the DC supply input should do the trick.  It will act as a power reserve for the brief period while the UPS is switching over.  4700µF should be adequate, and it wants to be rated at a minimum of 16V (12V PSU voltage plus some de-rating to cope with pulses and spikes from the switching):







simulate this circuit &ndash; Schematic created using CircuitLab

If you want to get really fancy you could build a little Pi filter, though it shouldn't be necessary:







simulate this circuit