Colonial missionaries and settlers move into Iroquois lands causing tensions. The Iroquois are forced to adopt more colonial lifestyles or cede lands in exchange for promises of respecting their slightly reduced territory, more settlers move in and the borders are moved again, repeat.
The British attempt to maintain good relations with the Iroquois due to their continued loyalty but the English colonists will always come first. Constant antagonism and demand to open the land on the colonists' part make the situation increasingly tense. Forward a few years until colonial population growth and power make it more feasible to remove them and then the colonial governments act (probably independently and against any orders from England) and destroy the Iroquois' ability to resist settlement or field any military force. There will probably be some resistance to turning on former allies from London but they'll cave to pressures from the colonial governments as fait accomplli.
I can't see the Iroquois surviving. In the long term there's no way London would continue to side with a comparatively weak native confederation against the growth of a profitable and successful colony of their own citizens and the colonists are strong enough to defeat the Iroquois on their own. If the Empire fails to constantly rebuff the colonials they'll find that the colonial citizens/government destroyed the the Iroquois while nobody was looking. All it would take was a few months of apathy by England for the colonists to antagonize them into starting a war and then take the land while claiming self defense.
The Americans mainly wanted the Ohio Valley. I don't think there was anything on Iroquois land that they wanted and the main reason people went there was because the revolution had left a power vacuum in the area. I'm not saying it will last forever but it might lead to a radically different reproach to American-Indian relations.
Not exactly a British legal system. New France got to keep their own system (and still does, Quebec uses civili law instead of common law).They'd need full British citizenship rights & defined, deeded ownership of lands - even if ownership were at tribal level; as a starting point.... Plus, they'd need a British modelled legal system of some sort; to settle land and criminal disputes.
All in all, it would have been tough sledding.