I went and sorted the rings by zone myself, that way it's easy to check whether you missed one or not when going through NG++
Figured it might be useful for others, too, because it seems efficient to just straight up speedrun through NG+ (while grabbing all 8 slabs) and ignoring all else, then playing through NG++ the normal way again.
+1 and +2 ring locations sorted by zone
==Cemetery of Ash==
Speckled Stoneplate Ring +1
Wolf Ring +2
==High Wall of Lothric==
Fleshbite Ring +1
Ring of the Evil Eye +2
==Undead Settlement==
Life Ring +1
Poisonbite Ring +1
Covetous Silver Serpent Ring +2
==Road of Sacrifices / Crucifixion Woods==
Lingering Dragoncrest Ring +1
Chloranthy Ring +2
==Cathedral of the Deep==
Ring of the Evil Eye +1
Ring of Favor +2
==Farron Keep==
Magic Stoneplate Ring +1
Wolf Ring +1
Dark Stoneplate Ring +2
==Catacombs of Carthus==
Thunder Stoneplate Ring +1
Ring of Steel Protection +2
==Smouldering Lake==
Bloodbite Ring +1
Flame Stoneplate Ring +2
==Irythill (pre-Pontiff)==
Chloranthy Ring +1
Covetous Gold Serpent Ring +1
Wood Grain Ring +2
==Irythill (post-Pontiff)==
Ring of Favor +1
Havel's Ring +2
==Irythill Dungeon==
Covetous Silver Serpent Ring +1
==Profaned Capital==
Flame Stoneplate Ring +1
Magic Stoneplate Ring +2
==Consumed King's Garden==
Wood Grain Ring +1
Sage Ring +2
==Untended Graves==
Ring of Steel Protection +1
Life Ring +3
==Lothric Castle / Dragon Barracks==
Life Ring +2
Dark Stoneplate Ring +1
Thunder Stoneplate Ring +2
==Grand Archives==
Sage Ring +1
Lingering Dragoncrest Ring +2
==Archdragon Peak==
Havel's Ring +1
Covetous Gold Serpent Ring +2