Hi, just purchased Oblivion GotY. When I save the game, it says "Save successful", but no save games are showing up except the single auto-save. I've looked through the game files but can't find anything. Any help would be appreciated.

I've got the same problem. Any ideas here?
I assume it's due to the fact that the save path contains a space character, which is "C:\Users\<MyUserName>\Documents\My Games\Oblivion". I did not set this path. It's been taken by GOG Galaxy (I guess).

Edit: I changed the save game location in the Windows registry (HKEY_LOCAL_MACHINE\SOFTWARE\WOW6432Node\GOG.com\Games\1458058109) but it did not change anything. That sucks. So I removed the game from GOG Galaxy and am currently downloading the offline version. I'll give it another try with that and let you know as soon as I have new information.

If you think it's Galaxy-related, you ought to report it to GOG support (don't expect them to act because of your post).
As to why others don't seem to have the issue: I'd guess that the number of Oblivion players using Galaxy is low. No surprise for me: At least the long-time GOG customers (which are also likely to be the majority of Oblivion players) prefer GOG over Steam because they don't like Steam's DRM and user / usage data collection processes. And Galaxy is too close to Steam to these people's liking.

FOUND A FIX FOR THIS (at least for me):

Had the exact same problem. Autosave worked but couldn't save manually. The game would report that saving was succesful but the save wasn't actually created.

In my case this was caused by an incompatibility between Oblivion and my Documents folder being auto-synced with Onedrive.

Here's how to do it:
-click on the Onedrive icon in the lower right corner of the screen
-click More -> Settings
-go to Backup -> Manage backup
-under Documents click Stop backup and confirm with another Stop backup

Now your Documents folder should be located at c:\Users\(username)\Documents instead of c:\Users\(username)\Onedrive\Documents AND Oblivion saving should work!