After a few days spent on looking all over the internet for the solution I just laid my hands on pirated Steam version and it worked right away. I just switched exe files and finally can play.
I encourage you to do the same or just buy the game on Steam.
It's really a shame that after all this time people are complaining GOG won't do anything.
It's likely one of the best, and for some the best SW game on their service yet stays unplayable for many. Can't GOG just ask Steam for their executable if they can't fix it themselves?
I just hope that people stumble on this answer and avoid similar misery I had to go through.
P.S. If that's the first answer on the subject you've found and you think that some other trick will work - no, it won't - new drivers, tuning settings, installing obscure patches... it just won't work in this case.