First I am going to try this with George III as king, since I think it would be a bit of a stretch to have Freddy live on another 9 years if he lived longer at all.
Let’s say the Albany Plan is approved with some modifications that make the resulting union explicitly subordinate to Parliament and the crown. Its implementation is suspended until the conclusion of peace with France, the crown and Parliament arguing it would be imprudent to attempt to select and install a new government in the colonies while they are locked in combat with their ancient enemy. The colonial delegates agree to this and the matter is settled, though in the dark days of Loudon and Braddock there are some calls for immediate implementation (these are few and fairly muted – most feel the reasoning behind postponement in sound).
The war progresses pretty much as OTL; when the news of the surrender of the French in September 1760, many in the colonies believe their time has come, but now there is a new question: do the conquered lands go along with the other colonies into the union? There is much debate about this, for at first Britain is unsure of whether Canada or the West Indies will be returned to France in the peace treaty. London uses this as an opportunity to stall, saying that with peace so close, it would be unwise to form a union with potential members remaining unaffiliated. This argument is accepted grudgingly.
On February 10th, 1763, the Peace of Paris is signed, giving Great Britain Canada and returning the West Indies to France. Immediately London is deluged with petitions and envoys requesting and almost demanding immediate implementation of the Albany Plan of Colonial Union. At this point there is a crisis in which the new king, George III, considers reneging on the deal, and some argue that things have changed so during the war and its aftermath the plan should be dismissed entirely. The colonial agents are outraged at such a suggestion and point out the changes in question were wrought at their hands and, besides, the Plan was given the royal assent in 1754, well before the supposed changes took place. Ultimately cooler heads, especially William Pitt, speaking in the Lords in one of his most famous speeches calling on his fellow peers to keep faith with their kinsmen across the sea who endured such hardships on behalf of the Empire and struck such a blow against “perfidious Gallia,” prevail and the moratorium on implementation is, finally, lifted. The union formally goes into effect on January 1st, 1764, almost a full decade after being first approved in Albany, NY.
The first President-General of the new Dominion of America is (for right now) Sir William Johnson, a loyal crown servant and expert on American affairs. The assemblies of the several colonies select from their own numbers the requisite delegates who gather in Philadelphia, that being the largest city in BNA at the time and also centrally located. Nova Scotia is included with three delegates and so is Canada with 2 delegates, both Frenchmen but harmless since they make up less than 4 percent of the Council and can be safely ignored (this state is not likely to continue as the population of Canada begins to grow significantly later in the 1700s and 1800s).
With an American local gov’t coming to Philadelphia, there is no Proclamation of 1763, since the settlement of new lands is covered by sections 12-14 of the Albany Plan.
12. That [the Grand Council] make all purchases from Indians, for the crown, of lands not now within the bounds of particular Colonies, or that shall not be within their bounds when some of them are reduced to more convenient dimensions.
13. That they make new settlements on such purchases, by granting lands in the King's name, reserving a quitrent to the crown for the use of the general treasury.
14. That they make laws for regulating and governing such new settlements, till the crown shall think fit to form them into particular governments.
The Royal Proclamation in OTL caused much resentment and could be considered the first proximate cause of the ARW. W/out it, things are much better, even if London is dragging its feet in putting the new colonial gov’t in place.
The new gov’t’s first test is to deal with Pontiac’s War. I am not totally sure how this plays out, though. In OTL it was put down by General Thomas Gage militarily and William Johnson diplomatically; in TTL after 1 January 1764 Johnson is P-G, so he and Gage could still work on solving the native problem similarly to OTL. Let’s say, just for fun, that the colonial gov’t and the British military establishment in NA work together extremely well, the new union basically vindicating itself and earning credibility in many eyes.