What I’ve just found out is if you delete some of your older saves, then auto-saves and regular saves work fine. Once I deleted out several of my older saved games, more than one auto-save would appear, and my manual saves also appeared. When there are too many saves (I didn’t note the number), the game will indicate it has saved successfully but it doesn’t.
Also, when you delete (it’s called Remove in the game) a saved game, it doesn’t get removed from the list immediately. But if you close the Load Game dialog, then open it again, it is removed. This is a simple issue of not updating the list after deleting the save.
I hope Owlcat investigates this (and I hope others in these forums do the same – I would be interested to see if the PS4 has the game issues) and fixes these problems.