Wire a relay so the normally-closed circuit sounds your alarm, as you have drawn.
A VERY time ago I wired a 240V DPDT AC relay among my household fuse boxes (following a little reorganisation of which fuseboxes fed where) - so that particular fusebox feeds all the lights, the central heating boiler electronics, and the circuit that feeds the TV etc. The relay is wired so that while the normal mains is working the relay feeds the mains straight through to that fusebox - but if the mains drops out, the relay also drops out and connects that fusebox to a 2KW generator instead.The circuit you posted would work fine for that.
I put a drop-out relay in an outlet strip once. You plug it in and press reset. When the power dropped out and came back the relay would stay de-energized. I'd plug an auto-answer modem into it. I could call the lab phone and if it didn't answer, we had a power failure.
I coould then goto work and reset the I-V-t overnight measurement.