A few minutes seems correct for one place I worked.
This was back in the 90's, before UPS technology got really interesting. Our system was two large rooms with racks and racks and racks of car batteries wired together. When the power went out, the batteries took over until the diesel generator could come online.
I saw it work during several hurricanes and other flood events.
I always found the idea of running an entire building off of car batteries amusing. The engineers didn't share my mirth.
The "when" shouldn't really matter- Diesel engines aren't a new thing. Warming them up isn't really a thing either- they'll have electric warmers hooked up to the building power to keep them ready to go.
Edit to add: I was at a place that took over a company that had one of these. With all of the dead batteries, it was just a really really large inverter taking the 3-phase AC to DC back to AC with a really nice and clean sine wave.