So, I just defeated one of the first bosses in the game, and it is my understanding that there is an achievement associated with this. Yet, I do not have any of them unlocked. It is my understanding that, so long as you play through the Galaxy Client and connected online, the achievements should be unlocked, as well as in game time reported. I am seeing the reported gameplay time, but as I said, no achievements have unlocked. Any ideas as to why?

Check in the Galaxy general settings that Achievements is indeed checked. Otherwise, there could be some kind of communication error between the game and Galaxy or some kind of delayed reaction. You should have received an achievement as early as for reaching the first gate.

(Sorry for bad English) Do you launch game from within GoG Galaxy, not just from desktop icon? Do you see an overlay message "Hellblade: Senua's Sacrifice now playing. Shift+Tab for more" during game startup? Does Shift-Tab and F-12 (screenshot) hotkeys are working?

I am launching the game through the Galaxy Client, the overlay is working and I can take screenshots, I have also double checked that achievements are enabled. Does the GalaxyCommunicationService need to have internet access to unlock/update achievements? I have a habit of not allowing internet access to a program unless I know why it needs to connect. When I originally installed the client, I only allowed a few Galaxy related files internet access with my firewall. I went through and allowed all of the Galaxy related files access, but for some reason, my firewall is still blocking the GalaxyCommunicationService. Also, yes, I am allowing the game itself access.

I should probably also mention that cloud saves are not syncing and the pane on the right side of the overlay is blank, then loads a "not connected" message after a short delay.

Can't say for sure, but I've found this https://www.gog.com/forum/general/offline_achievements/post5 and this https://www.gog.com/forum/general/offline_achievements/post9 so probably you don't have to be online, just logged in.

Thanks for those links. I did some more research on the issue and it seeems other people have problems unlocking achievements in various other games as well, so maybe its just a bug that needs to be worked out. I also forgot to mention that occasionally upon starting the game, I get an error related to Galaxy64.dll, so maybe that may be related. I have seen other people mentioning issues related to the .dll, as well.

Honestly, restricting galaxy files from internet connection is probably causing you, and others, more disservice than helping.

You don't even need to be online to run the games, and galaxy is optional. It only reports stats if you let it. Why restrict it?

The best comparison I can give is when I was 23 and I did a fresh install of Windows 2000 Pro. I wanted to reinstall Lineage 2, but I was super paranoid about the government spying on my computer. So much so, that I installed Spybot S&D immediately after installing Windows, and as I tried to install DX8 to play Lineage 2, I accidentally blocked a DirectX runtime from installing, basically fubarring the whole install. That mistake cost me several hours to figure out, because I was in a deep psychotic break at the time and was convinced the file was a spy program sent by the government because I had torrented some music 6 months before.

My point is, there IS such a thing as being too careful. You guys who want absolute control over everything on your system are commendable, but if you keep turning stuff off, it IS going to break stuff.

I have the same problem.
The achievements unlocked normally through most of the game, but after the Worrior achievement to the end of the game, none other have unlocked.
I have always launched the game through Galaxy and haven't changed any of the online settings.