Akarianen: I think I have found the solution to this problem. It seems, that for some reasons, firewall was blocking some of GoG functions (although, usually applications are asking for adding them to the whitelist, when they are trying to send/receive some traffic and firewall is preventing them to do so).
AriesCZ: Sadly, this hasn't helped my problem, so it is obviously something more.
I have whitelisted all 4 files in my firewall, did several reboots, played for extended periods of time, but still nothing...
I am glad you managed to resolve the issue at least for you though...
Only two things are comming to my mind now, that may help:
First, would be just to disable firewall (for just few minutes, for testing purposes) and run game from GoG Galaxy. Your achievements should be loaded as the game starts (and maybe load game from your last save, but I think there is no need to do more). Give it, lets say, 5 minutes, than look at the GoG Galaxy, both on the general tab of the game and the Progress tab, where the achievements are listed. If something new will appear, it means, firewall was somehow still blocking something. Maybe check once more if the both incomming and outgoing traffic is allowed for files mentioned in previous post or search if more of them are needed? But you will at least have a lead.
Second, would be to upgrade the GoG Galaxy version. I had problems with it, as the auto updater was stuck, looking for update forever, so I have decided to just download installer and run it (as was proposed in some forum post). It did not made any difference, as my version was the newest, but I guess trying that step will not hurt, maybe something was refreshed in the background?
You can also try to just reinstall GoG Galaxy at all, uninstall, clean registry with CCleaner or sth and install it again, from zero.
Aside of that, I don't recall doing anything more to solve this problem.