[PROPOSED SOLUTION - confirmation needed]
Guys, I probably figured it out! And yes, we were most probably just dumb 
.
The key is in the checklist…from cold & dark, you are turning Battery on, then External power. And then there is step “Engine generators: ON/Fault displayed”…what I was doing in this step: I clicked on the both Engine Generators buttons to “turn it ON” as I thought (so “Fault” message dissappeared). But this is wrong. Engine generators are already ON, but engines are not running in this stage yet, that’s why it’s in Fault of course.
So basically you should leave it in “Fault” and continue with next step (AI will do same thing, if you will let checklist on AI, as I tested).
So basically cause of this issue was, that we turned the engine generators OFF and then shutdown the APU after engine start. So basically from that moment, we were flying just on battery power. And I believe, those ~2h means, that battery were completely drained out and aircraft lost all it’s power (engines running, but those were not producing any electric power, because generators were off).