This scenario gives rise to a whole lot of interesting and intriguing possibilities.
My POD is 1782. The British government - either that of the Marquis of Rockingham or the Earl of Shelburne - gives each of the 13 American colonies their independence as separate states.
According to their founding charters the land grants for aome of the colonies stretched from the Atlantic Ocean to the Pacific Ocean. When they became independent they would want to expand into and annex land to their west. There would also be conflicts between the states. So Pennsylvania, Virginia (which then included OTL West Virginia), and Connecticut which nominally extended to the Pacific Ocean, would be in conflict over OTL Ohio; Virginia would expand into OTL Kentucky; North Carolina into OTL Tennessee; and Georgia into Alabama and Mississippi.
The northern states would want access to the Great Lakes which would be another source of conflict.
The smaller states would most likely combine with, or be annexed by, the larger states. I don't see Delaware or Rhode Island still in existence as independent nation states in 1882.
It is possible, perhaps probable, that some time after 1800 settlers west of the Appalachians, and north and west of the Ohio River would declare their own imdependent states.
In this scenario there would be no Louisiana Purchase, unless acquired by one of the states. A feasible possibility woould be by Jefferson as President of an expanding Virginia. If not, then assuming that the Napoleon Wars happen as in OTL, then I would guess that the Louisiana territory would be divided between Britain and Spain at the Congress of Vienna in 1815.
Britain would claim all of the Oregon Territory, which would later become part of Canada. Also possible British purchase of Alaska from Russia, with Hawaii as a British protectorate.