It's actually a safety feature that the APC does NOT resume after a power outage (your food could have been in the danger zone for too long and be harmful) - you would be unaware and potentially get a nasty dose of food poisoning.
The way to mitigate this is with a UPS (uninterruptible power supply). If you want one that will keep the APC running for a prolonged period, you want one that supports about 1500kVa.
Anova would have to change the internals of their appliances to make them aware of a power outage and notify their cloud when this happens (amongst other things, they'd have to put a battery on their system board) - just about the only way that you could keep from having false positives. (don't want people throwing their food out because someone turned off their wireless access point or modem).
The app is the same for all of Anova's appliances - both those supporting only bluetooth as well as the wifi/bluetooth APC.
Edit: One more thing that can help mitigate the loss of power - if you use an insulated vessel (such as a cooler), that is also covered, they lose heat very slowly - so it may keep your food from entering the danger zone before you notice the cook has stopped.
And, yes, Anova could provide a notification whenever the cloud connectivity is lost for the Wifi models - not the bluetooth ones such as the Nano though - they don't have a cloud connection on their own.