Perhaps it's because of my obsessive-compulsive tendencies, but I still want to delete all the old cloud saves. CDPR doesn't provide any way to delete these four saves (there's also a "can't turn back" save).
I get what you're asking, but it's really tricky to do, and exactly what steps need to be taken will be different whether you're on Steam, GOG, Epic, Xbox, Game Pass, Playstation, Nintendo, etc. What
@LeKill3rFou says is true -- you can freely start new games without worrying about the older cloud saves interfering.
I know I've done it for Steam games, but it's been a looong while since I tried it. Not even sure if it works the same way, anymore. It was a matter of disabling cloud saving altogether. Then, you have to find the local files on your computer that sync with the cloud. These are usually the save files that are in the AppData folders on PC. Those needed to be removed from the folder, then you had to go into the game an begin a new playthrough or load the local game you wanted to keep. Lastly, you have to re-enable cloud saving so that it syncs with the now re-built save folder, which will ask you if you want to restore or permanently delete the save files that were previously synched. Delete them, and the new sync will include only the new files.
As stated, that's a lot of hassle, no guarantee that it will work that way anymore, and (based on what I was just reading) you may actually need to request account action from the Support teams on whatever platform you're on. Protecting cloud saves from outside actions to avoid attacks and identity theft is kind of the whole idea of cloud security.
If what I've explained doesn't do it, you'll need to directly contact the platform you're on and ask how. If cross-platform saves were made, you may need to contact each platform in turn. (It's also possible that cross-platform saves may automatically restore cloud saves on the other platforms they're linked to.

) So...good luck with it!