I just finished Tyranny and during my first couple of playthroughs I did not get any achievements either. I only started unlocking them once the following things were all true:
- start the game from within GoG Galaxy
- "Cloud saves" and "Overlay" are enabled (Game Settings > Manage installation > Configure > Features)
- in Galaxy's General settings, all boxes under "Game features" are checked
- be online while playing; sometimes I also had to restart my router to force a new connection to my ISP or Galaxy would refuse to connect to GoG's services, telling me I was offline (which I was not)
At some time I just gave up and did all of these. Someone with more patience might be able to give a more precise response.