A few things that stood out to me, one, Eitiri still had his hands, two, Tony repeated one of his big lines from Age of Ultron ("How can we put a shield around the world."), three, Gamora is wearing the armor she wore while with Thanos, and not the sleeker outfit she wore while with Rhonin. This makes me think the divergent event had to happen before 2014, since Gamora never joined the GOTG, Tony never said the armor line (Otherwise why repeat it like you never said it before? Would he want to say it again after Ultron?), and the other realms abandoned by Loki after Dark World were not abandoned long enough for Thanos to cut off Eitiri's hands.
My theory? When Tony redirected the nuke at the end of Avengers and launched it through the portal, instead of falling back down to Earth, he got trapped, and drifted through a wormhole, ending up on Sakaar. This is why the armor has Sakaar symbols and seems to be battle ready for hand-to-hand combat, likely having taken part in the various arena battles, and probably over time even became the Grandmaster's Champion. Thanos learns about this sometime around the events of GOTG, and rather then sending his daughters to help Rhonin, he instead sends Gamora and Nebula to capture Tony. They go to Sakaar, and likely get trapped in the same games, all the while trying to get close to Tony, but the sisters bond and work out their differences during these events. When they finally get to Tony and free him, neither sister wants to follow Thanos anymore, and they decide the best way to stop him would be to prevent him from ever getting the Infinity Stones by destroying them. With Tony, they escape Sakaar and head to Nidavellir where Tony works with Eitiri to create the Infinity Crusher, which will be able to destroy the stones, but only when they are all gathered together, using the Soul Stone as a power source. Not long after this, Thanos attacked Nidavellir and forces the dwarves to start making the Infinity Gauntlet. Now who collects the stones, either the trio (Tony, Gamora, Nebula) or Thanos, I can't really parse, but I know it would end with Gamora, having known for a long time the Soul Stone was on Vormir, heading there to collect it with Tony and Nebula, only to run into the sacrifice requirement. Nebula ends up sacrificing herself to a tearful Gamora, to help them get the stone. It all ends with the stones together, Thanos appears and attempts to take them, only for Gamora to use the Infinity Crusher, destroying the stones and likely killing Thanos in the backlash. Free of him, but still torn because of their weird relationship, she takes his sword as a trophy / reminder. The scene in episode 9 on Nidavellir is Tony and Gamora returning to free the dwarves, but also destroy the recently finished but unused Infinity Gauntlet.