At the time of the Revolutionary War, the Oneida had increasing social and economic ties to the colonists. The Oneida lived closer to the colonists and, compared to the other nations of the Haudenosaunee, American missionaries had far more success gaining converts among them. By and large, the same can also be said for the Tuscarora, but at the time, they weren't fully fledged members of the Haudenosaunee just yet. During Grand Council meetings, the Tuscarora representatives were allowed to speak and make arguments in favor of their particular stance - supporting the colonists in this case - but they weren't able to vote.
So the primarily Oneida-based pro-colonist faction favored peace, if not outright support with the colonists, in order to maintain their established connections and to avoid becoming the front line in a war against them. Meanwhile the anti-colonial faction, with particularly vocal proponents among the Seneca and Mohawk, were concerned about further intrusion by the Americans into lands claimed by the Haudenosaunee.
When the factions failed to reach a compromise, the council fire at Onondaga was extinguished, and the confederacy of formed by the Five Nations came to an end. After the war, there were two official successor confederacies, plus a third unofficial one. The two official successor were the Six Nations of Grand River in Ontario and the Council of Buffalo Creek in western New York. Both kindled new Grand Council Fires and split the wampum belts belonging to the 50 lineages of Grand Councilors between them. The Six Nations, a the name implies, finally elevated the Tuscarora to full member status. In 1924, Grand River abandoned the hereditary title system for the its Grand Council and replaced them with elected representatives, which is the system still in use today.
The Council of Buffalo Creek disbanded in the 1840s, over a disagreement concerning the sale of Buffalo Creek to the Americans. In 1847, the Council Fire was rekindled at Onondaga for the first time since the Revolutionary War, when the 14 Onondaga councilors from the now-defunct Buffalo Creek council met on their own. The modern Tadodaho (formerly the prime minister of sorts of the whole Haudenosaunee) belongs to this council.
The unofficial successor to the Haudenosaunee were the Seneca and Cayuga who lived in the Ohio Valley at the time of the Revolutionary War. They didn't inherit any share of the wampum belts and didn't form their own Grand Council in the traditional sense. Instead, they joined up with the newly formed Western Confederacy, a pan-Indian confederacy based in what's now Upper Sandusky, Ohio. While the Western Confederacy was spearheaded by Iroquoian communities - in this case, the Wyandotte rather than any of the old Haudenosaunee nations - most of its members were Algonquian, including the Shawnee, Delaware, and Miami. The Western Confederacy broke up after the Treaty of Greenville in 1795, and the Seneca and Cayuga portions of it eventually became the Seneca-Cayuga Nation in Oklahoma during the later Indian Removal period.
I have not heard this before. Do you have sources? I have heard of the main Iroquois nation's conflict with the Huron, who are culturally the same but politically allied with the Algonquin nations.