I have been analyzing the issues I described in the "Achievements not popping" thread, and with a lot of logging and experiments, I have discovered what goes on here.
Yooka-Laylee has a built in achievement system, which registers unlocked achievements in the profile.dat file, which is stored in the users AppData area.
This works just fine when installing and using the game without GOG Galaxy, but when running the game with GOG Galaxy installed, it seems like GOG takes over the tracking of achievements, which will force the game to cancel all achievements that were unlocked before GOG Galaxy was installed and any changes to the settings will cause the before mentioned profile.dat file to be overwritten with the GOG tracked achievements.

Fortunately I had a backup of my saved games and profile for the game, but restoring that did not fix the achievement problem, as the game launches GalaxyCommunication.exe on start, which overrides the profile.dat again.
As long as GalaxyCommunication.exe still exists in the ProgramData GOG.com sub folder, it will be launched and cause this problem, as it syncs with GOG Galaxy online and recreates the online achievements every time.
I even found the files that are synced and renamed the folder for it, which makes my achievements work again as long as I am offline, which causes GalaxyCommunication.exe to create a new folder, but as soon as I go online again, it overwrites my data again.

To try and find a fix for this I even tried disabling all the features in GOG Galaxy, but this did not fix the problem either, which I would consider a bug in the system. Disabling cloud saves and/or achievements should be a valid way to fix the problem, but it is not the case :(

At this point, if I want to keep my achievements in the game, my only option is to uninstall GOG Galaxy and remove the "GOG.com" folder from my ProgramData manually as this is not always removed by uninstalling GOG Galaxy, which I would also consider a bug. It seems GalaxyCommunication.exe keeps running for a while, even after closing games and the client, which causes the uninstallation to be incomplete, so it might be a good idea to check for this after uninstalling.

It is unclear which of these problems are caused by GOG and which are by Playtonic Games, so I think this forum is the right place to put this. If any of these companies read this, please respond so I know my analysis is not in vain.

For other users, I hope this gives you a chance to avoid or fix the problems, so you won't have to loose achievements when playing offline.

I have now updated my game to version 1.08, but unfortunately the problems with GOG Galaxy integration has only gotten worse. Now the game freezes on exit and it still overwrites my achievements, even though I've tried turning off all Auto-Updates, Cloud saves and In-game overlay in the settings for the game in GOG Galaxy, and even tried turning off the global toggles (Auto-update games, Cloud saves, Achievements, Game time tracking and Overlay).

I would expect GalaxyCommunication.exe to leave the game alone with these settings, but the only way to avoid having my stats and achievements destroyed is disabling the GalaxyCommunication service and run the game directly from the YookaLaylee64.exe file. I accidentally ran the game from the shortcut, which opens GOG Galaxy, but then the service is reenabled by GOG Galaxy and the problem is back.

Anyway, this is the only way for me to enjoy the game properly at this point to avoid crashes and lost progress, and I am sure I am not alone, so I hope you guys can use my workaround for these problems:

1) Exit from GOG Galaxy completely by clicking the "gog com" logo and select "Exit".
2) Go to the start menu and select the Run or search input field.
3) Type "services.msc" and press enter.
4) Locate "GalaxyCommunication" on the list of services and double click it.
5) Set "Startup type" to "Disabled" and click "OK".
6) Run the game directly from YookaLaylee64.exe in the game installation directory if you installed through GOG Galaxy, or just run the game from the shortcut if you installed it manually.

Yes, I will read up on the proper way to submit bug reports there soon. A fix is always better than a workaround solution.
Thanks for the heads up :)