Early on in the ARW, Britain did negotiate with Russia for troops. Suppose they were actually able to convince Catherine II to formally support Britain, would it have any effect on the war or the founding of the United States?
Personally, I doubt that the Russians would be enough to win the war for Britain but assuming that some of the Russians defect like the Hessians did it could be enough to found a Russian ethnic community in the United States or having an experience fighting the Russians could impact American foreign policy, assuming that the Americans hold a grudge.