You can go from cold bird to flying in about 90 seconds. It's all a matter of timing. The doppler nav will come up about 30 seconds after that, but it's not critical for flight and initializes just fine while in motion. I think the ABRIS may still be warming up, too, but rest of the systems will be up and fine.
Of course, a real Ka-50 would never be started up that quick - the INS takes minutes to align, not instantaneous like in the sim, and no sane pilot would put that stress on the APU and engines. But it could be done in the sim.
Here's an optimized scramble start list I made back in the day. Should still apply to Black Shark 2.