Electronic – arduino – 5V Battery Backup Supply for GPS

This is more of a comment than an answer because your requirements are made fuzzy by your illogical (don't take it to heart) circuit:

Are you looking for this type of basic configuration of your batteries: -

If so, then add the relay contact either in series with the 12V battery or in series with Vout. I'm assuming that your arduino will always take power from one of the supplies. I guess you'll need some kind of charge circuit for the 12V to keep it topped up too. This can be done with a schottky diode in series with a resistor from 13.8V directly to 12V BUT it depends on your battery type and I'm no expert on battery chargers.

To answer this a bit clearer, when the charging circuit (aka rectifying the mains, power to recharge) goes out. It counts as a "closed circuit". Yes you can add measures and all, problem is, it may drain your resources. Deep cycle batteries can help cause they slowly trickle power, enough to run your router. I am running a little bit of sleep. So no need of special circuitry to switcj from charge to drain. However if that becomes an issue, just have an off on switch on the discharge ONLY. Otherwise it can charge your battery and power your router if assuming the circuit is parallel. Hopefully it makes sense!