I spent more time on it and found the reason for my issue.
To start the engine the plane must be throttled down. The throttle doesn't move (neither in the cockpit or on cotrols indicator (RCtrl + Enter)) before engine is running. That's why I haven't noticed it isn't throttled down. Recently I bought new pedals and DCS assigned one of their axis to throttle. I missed it, so I couldn't start the engines.
I guess the solution is check your binds before posting on the forum.