Though on the Greek side it's even murkier.
Kratos explicitly states that the Thread of Fate span all life and time, and he manages to travel back to the past of the Underworld using his thread alone, to Tartarus, to teleport the Titans.
However, he only succeeds in changing the past of the Mortal World, but the Underworld's past does not change when he does something of an act like this, nor does it change the past of the Mortal World (The Great War forged the landscape of the Mortal World from Tartarus below), worse still, it also does not even change the future of the Underworld as the plaque outside Tartarus still notes of the overwhelming Olympian victory over the Titans (If they didn't win, Thera would never be there to give Kratos the Bane, and Poseidon is visibly pissed about Atlantis's destruction, Persephone is still very ******* dead, Helios remembers Kratos saving him from Atlas and Hades remembers Kratos killing Persephone).