What if Catherine the Great had agreed to supply troops to the British?
At the very beginning of the American Revolutionary War the British at first did not want to admit that the situation in America was as serious as reports made it because they would then have to send large forces to America with all the expense and political fallout that entailed. The news of the Battle of Bunker Hill, which arrived in England on July 25th, following on the news of the rebels’ seizure of Forts Ticonderoga and Crown Point, dashed their hopes for a quick and easy resolution of the problem; it was clear a large army would be needed.
General Gage’s estimate of 20,000 needed for the job was taken as accurate; with only 27,000 troops in the British Isles and the dominions clearly more needed to be raised. Recruitment was forecast to raise this to no more than 32,000, and at least 50,000 would be needed for the expeditionary force and to meet other requirements, so the British began negotiations with foreign governments to hire additional troops.
At this point Catherine the Great offered 20,000 Russian troops. By the time the cabinet considered the offer and formed a reply she had reconsidered and withdrawn it, partly because Pugachev’s Rebellion, only recently put down (1774), had made her nervous and the Russians had just finished the First Russo-Turkish War (1774 again), so needed to catch their breath, so to speak. The disappointed British then turned to the German principalities and hired nearly 30,000 of their troops to fill the need.
So, assume that Pugachev’s Rebellion is less successful and the Turks are more easily defeated, so that Catherine is more confident and reaches an agreement with the British to supply 20,000 troops. Clearly they cannot be available before 1776; by the time agreement is reached and the troops are ready it will be November or December 1775 and the Baltic ports will be frozen over. So they will be sent in May or June of 1776, arriving in the British Isles by July or August.
Given that, the British would hire German troops anyway, so will have nearly 75,000 troops on hand by August 1776, of which some 55,000 could be sent to America to add to the 10,000 already there. (Troops on hand: 32,000 British and Scots, 20,000 Russian, 23,000 German. Requirements: Ireland 7,000, Scotland 3,000, England 10,000.) With this large a force could the British finish the job and crush the rebellion before the French stick their noses in?