An update on this. I started a new character and then the quest log updates normally (for the new character).
I tried uninstaliing the game and reinstalling it and transferring the save files but the result is the same. The problem turns out is even worse because not only the quest log is not updated, but the character doesn't get any XP from the quests either. So although I'm going through the storyline normally, the game engine fails to recognise neither that I'm receiving nor that I'm concluding quests. No wonder I got the stronghold and I was still level 1...
So, eventually I just started a new game with the same character build and game choices and did everything from scratch...At least I was not too far into the game...But definitely a very annoying, if not game breaking eventually, bug!