It seams that the bug fixed by itself. Today I tried to open the website and everything was just fine – successful update to 3.5.2 and all working. I haven’t done anything really, it just happened. Now I made my backup in case the problem reappears.
Don’t know what was the problem really, maybe the background updating takes more time than it says, I don’t know.
So, this thread could be marked as RESOLVED, but who knows what actually resolved it?
For others who experience the same issue I would recommend to wait 24 hours before posting their threads as the update might need more time than expected.