The thing is that Archolos isn't actually a DLC for Gothic, but a full conversion, separate, it's only a GOG requirement to have the game in order to download it (for free), doesn't require the game to actually play.
Those issues may be due to old browser versions? Or some privacy settings, maybe.

Otherwise, I'd still be curious what was actually left on the disk in that computer, as opposed to what shows up as installed, especially if you didn't somehow install the games under Program Files.

And... You seem in a state of panic at the moment, which is understandable under the circumstances but leads to a string of rushed and bad decisions that tend to only worsen the situation. Reminds me of what I did when there was that HDD failure in 2007, when on top of basically running around like a headless chicken, after "reviving" the drive enough to do so, putting it in another computer, I simply forgot to save some files that were... priceless, for personal reasons. And also made a bad choice about not backing something else up, at least on DVDs or something if I had no space otherwise.
At least learned from that to always have backups, multiple and regular, and also something of a checklist in case of a repeat of the issue (which there was, in 2012, when I didn't lose anything as a result).

I'm sorry to read this tale of tragedy mate. I guess there's a lot of easy advice that "you could have done this or that" that we could give you and that won't help you at all now (tho they might be useful in a future pickle).

BUT - the one advice I can give that might still be useful is reinforcing what Cavalary posted and I copied below:

I somewhat disagree with Cavalary that it is a longshot, because the few files you need (your actual savegames) aren't really that big. I can't be sure of the odds because it depends a lot on how big is your HD and how much space is currently used, but those files are tiny; it is possible the new install just overwrote a different area of the hard disk.

Besides, even if your last save got overwritten you could still recover a previous one. Getting a save 90-hours in isn't as good as recovering the 100-hours save but it's still much better than restarting from scratch so the chances that there's still something worth recovering aren't too bad. But he is right when he says that the more you use your laptop now the lower your chances get.

However, the problem with these disk recovery programs is that they are quite confusing and ridden with tech lingo that even people that are used to tech stuff barely ever heard. I never saw one of these programs that was even slightly user-friendly. If you have some friend with the know-how to handle low level technical stuff this is the time to call them.