Assuming the revolution gets that far. The appointment of Washington was instrumental to getting the South to go along with the North.
Agreed. Washington's early death won't just effect the Revolution, it could very well effect the North American portion of the Seven Years War too. Let me explain.
Washington's two pre-war missions to the Ohio valley region were purely colonial affairs. With a few royal charters in hand, Virginia had formed a company which made several land claims in the Ohio Valley and her political leaders were heavy investors in that company. Sporadic settlement and trading by the colonists had finally triggered a French military expedition to reaffirm French claims to the region and punish tribes which had been dealing with the colonists.
When Virginia's governor, Dinwiddie, received reports about a newly constructed French fort, he dispatched Washington with a small party to deliver a letter demanding that the French withdraw. Washington delivered the letter, received the French commander's polite, but negative, response, and returned to Virginia. There he told Dinwiddie that the French had swept clean the region and intended to build a fort at Three Rivers (our Pittsburgh).
Adding the confusion, Dinwiddie had, before Washington returned to Virginia, dispatched a small party of militia under command of a man named Trent with orders to build a fort at Pittsburgh before the French could. The Trent party reached Pittsburgh and had just begun construction when a larger French force showed up and politely showed to colonists off. After hearing Washington's report and unaware that Trent's men had withdrawn, Dinwiddie sent Washington out again with reinforcements for Trent.
Washington marched out, heard about Trent's retreat, parlayed with a few natives who suggested their leader was unhappy with the French presence, and decided to push on. That's when the entire ambush/misunderstanding, withdrawal to Fort Necessity, and subsequent surrender took place and that's when I assume the OP has Washington killed.
You can see that the entire confused affair with triggered by the colonists and involved purely colonial issues. More importantly, the region being contested was of little strategic benefit to either side.
Unlike the French forts and positions in northern and western New York, the Maritimes, and along the Great Lakes, the French fort at Duquesne/Pittsburgh posed no real threat to the colonies. The Ohio valley was rather distant from both French Canada and Britain's seaboard colonies and would make for a very poor invasion route between the two. The strategic center of the war was in northern and western New York and the Maritimes as those regions were invasion routes.
Britain targeted Fort Duquense in 1755 because of Virginia's lobbying. The humiliation of her militia in 1754, not to mention land sales, had Virginia eager to evict the French from the Ohio valley. Britain, who needed to active participation of her largest and richest North American colony had to agree to an expedition against Fort Duquesne. Braddock was dispatched and the rest we all know.
After the Braddock disaster, Duquesne assumed assumed a status out of proportion to it's actual utility in prosecuting the war. Troops and supplies which would have been better utilized in New York were instead frittered away in Virginia and Pennsylvania against Duquesne. Amherst, the man who actually won the war in North America while Wolfe gained the glory, was only one of a great many voices who spoke out repeatedly against this fixation on Duquesne and the eventually successful effort led by Forbes in 1758 was going to be the last attempted no matter if the fort was taken or not.
If Virginia had been undeniably repulsed, instead of merely humiliated, the year before the war, Britain and her colonies may not have been as fixated on Fort Duquense as they were in the OTL. Braddock and his 2,000 men may have been dispatched to northern or western New York while the militia in Virginia and Pennsylvania concentrated on manning frontier forts.
More troops and supplies in New York earlier may have led to earlier British successes on that front. More militia and forts along the frontier, coupled with a lack of British expeditions stirring up the natives, may keep more colonists along the frontiers in their homes. In the OTL, there was a huge refugee crisis of sorts as most people along the frontier headed back east.
The death of Washington and the reasons why he dies are going to lead to immediate butterflies effecting a rather important war.