Pilot reported alternator failure during IFR training flight over terrain. The flight crew diverted to make a precautionary landing.

Narrative

During an instrument instruction flight in IMC with Person A (private pilot) at the controls in a Cessna 172; after having completed a practice approach to ZZZ to a missed approach (planned as such); we were getting vectors to ZZZ1; expecting the RNAV XX. Over the mountains; the fuel gauge began to indicate losing fuel rapidly; a low fuel warning light activated; and moments later we lost all Nav/Com/transponder/ADS-B out. I took over flying duties while a back seat passenger Person B (Commercial/Instrument rated pilot) accessed his tablet with Foreflight while Person A tried to diagnose the issue; including reducing electrical load and attempting to regain power. Believing that we had lost our alternator and had been on ship's battery power; I made the decision that we were in an emergency situation and asked Person B to provide vectors and safe altitudes based on his GPS equipped tablet. Person B accessed his hand-held transceiver while I continued to fly using the Garmin GI275 w/ battery back-up. Not knowing how long the battery back-up had been keeping the AI and DG operating; nor how long or how accurately I could trust the TC; I decided that we needed to get down and to ZZZ1 as soon as possible. Person A established contact on the handheld; [requested priority handling] (per my instruction as PIC); and provided position & altitude information based on his tablet. Following roughly; but offset from; the RNAV XX approach; we descended towards ZZZ2; making visual contact with the ground several miles out from ZZZ1. I saw Runway XX from a bit over a mile while at Traffic Pattern Altitude. The flight ended with a normal landing and long role-out. I believe I did all the right things given what we knew at the time. I believe we did a fair job of using all available resources; including cockpit resource management between the three of us. We checked the battery voltage on the ground; it had gone down to ~15 volts (nominal 24 volts). The maintenance shop is still looking to find out what happened to ship's power and why we never saw a 'volts' warning light when (putatively) we lost the alternator / charging system I highly recommend all flights into IMC carry tablets with GPS and hand-held radios. What could we have done better? There was a Davtron clock w/ OAT & voltmeter. Since we were in IMC; we had to on the OAT. We could have occasionally checked the voltmeter.