The biggest change for Indians comes after the US loses the "War of Secession." This happens before the Great War trilogy, which I haven't read. The Wikipedia entry seems to speculate what might've happened between books.
1862–1881: American Changes
In addition, the USA may have had a significant and victorious Indian War prior to the Second Mexican War, suggesting that massive firepower and a national need for some kind of martial victory propelled the United States to a much quicker and clear-cut answer to their Native American question, namely the extermination or deportation of the Natives.
The US also loses the Second Mexican War, with the following consequences:
1881–1882: The Second War Between the States (Second Mexican War)
The defeated United States, realizing it needs powerful allies to counter the Confederate alliances with Britain and France, begins an alliance with the German Empire and adopts many of its military and economic practices.
One of the few victories for the United States in the war, a battle in the Montana Territory against the British, produces two U.S. heroes who will be rivals for the next forty years: Brigadier General George Armstrong Custer and Theodore Roosevelt, Colonel of the Unauthorized Regiment.
Witnessing the collapse of the Republican Party, former President Abraham Lincoln, now an orator, allies with U.S. socialists and leads left-wing Republicans into their fledgling Socialist party. The Republicans soon begin a long descent into the general obscurity of becoming a solely Midwestern regional party, never again winning the presidency or a Congressional majority, losing around a third of its supporters to the Democrats, and all of its left-wingers (around one third) to the Socialists.
In the Great War between the United States and the Confederate States, the US overruns and eventually wins the state of Sequoyah. The US considers Sequoyah a territory and doesn't admit it as a state.
The CSA's defeat leads to the rise of the Freedom Party, a Nazi-like organization led by Jake Featherston. Featherston demands the return of CSA territory, including Sequoyah. In a 1941 plebiscite, Sequoyah is the only territory that votes to remain with the US. The explanation is that Sequoyah's white settlers outvoted the Indians, who would've preferred the CSA.
Mixed results for Indians
As always in alternate histories, it's fascinating to see what might've happened to the Indians.
On the negative side, there's the whole extermination thing. But this scenario--that the US Army would've wiped out the Indians earlier and more easily--seems questionable to me. It's just as easy to imagine that the US would've been too weak, distracted, and demoralized to pursue the Indians. The US might've left them alone long enough to organize an effective resistance.
On the positive side, the CSA created Sequoyah, a state primarily for Indians. And the US kept Sequoyah intact, even though it may have let whites overrun it. So the Indians did better in some cases and worse in others.
The overall message of this and other alternate histories is that Native defeat wasn't inevitable. Most alternate histories posit that if the Euro-American powers had been more divided, the Indians would've done better. Instead of fighting a unified America, the Indians could've helped themselves during a series of internecine wars. While the Americans were killing each other, they could've established one or more Indian nations in the unsettled West.
A key point is how easily this timeline could've occurred. All it took was one piece of paper not falling into the wrong hands. In the real world, the Aztecs almost killed Cortés, the Founding Fathers almost created an Indian state, Andrew Jackson almost died in a duel, and Cherokee relocation won by only one vote in Congress. Native defeat was a lot less inevitable than most people realize.
As in many of his books, Turtledove builds Timeline-191 by shifting between a large cast of characters. We see the changed history through many eyes and slowly learn how it differs from our reality. Some critics say these books have too much characterization and not enough plot, but not me. I love the rich blend of history and fiction in these books.
Turtledove's invented history seems totally plausible to me. There hasn't been one moment when I've said, "No, I don't believe that could've happened." When you consider that this series runs 11 books in all, it's a masterful achievement. If there's a longer alternate history with more basis in fact, I don't what it is.
Rob's ratings for the books I've read:
How Few Remain--8.5
Blood and Iron--8.0
The Center Cannot Hold--8.0
The Victorious Opposition--8.0
For more on the subject, see Indians in Turtledove's Worldwar and Was Native Defeat Inevitable?