Hey, save the Italy nonsense for another thread, guys.
Let's see if we can make something of this...
I'll make the horseshoe nail the attack on Quebec City. Instead, General Montgomery pulls back into New York. We'll say the ripples don't affect Washington's capture of Boston and the subsequent New York campaign, and General Burgoyne still proposes his pincer movement along the Hudson. But instead of Gates, Montgomery gets put in charge of the rebel forces in upper New York, and manages to beat Burgoyne somewhere around Albany. (This might be a bit random, but we'll put it down to the ripples--the rebels have a few more troops than they really did, and Montgomery is a better general than Gates.)
There was a rebel delegation in France at the time--actually led by none other than Ben Franklin. He gets word of the victory at Albany, and this convinces Louis XVI that maybe the rebels have a chance, so that declares war on Britain. (I'm not sure what the consequences of this will be in Europe, but let's leave that aside for a while.)
The French army and navy might be able to help the rebels directly, or maybe they just distract the British. In any case, this gives the rebels some breathing space. I think they may reshuffle their command structure a bit--Washington might have been popular with his troops, but he wasn't winning battles. Maybe they kick him upstairs, make him "Minister of War" or something, and put someone else in field command
From there, we need some luck for the rebels/French, but let's say they make it. London is forced to recognize the colonies' independence--maybe they just abandon them to focus on the war in Europe. (Anyone have any thoughts on how that might work out, by the way?)
At this point, the "Continental Congress" somehow has to turn itself into a real government, encompassing thirteen rather diverse colonies. At a guess, we see a power struggle between the radicals, wanting some sort of central state, and moderates wanting the colonies (or states, I guess) to go their separate ways. The balance of power will rest with the Army--usually a sign that a coup is coming. Let's say that Washington ends up as a figurehead for some sort of pan-colonial cabal. I don't see him being a better ruler than he was a general, so the "American Republic" will probably be gone within a decade. Instead, we'll get a northern and a southern federation, centered on Massachusetts and Virginia respectively.
From there, we can go anywhere. If Britain doesn't lose the European War, they'll be in a good position to reassert control somewhere down the line, possibly playing the states off against each other. Otherwise, maybe they fall into the French (!) sphere. Either way, eastern North America is not going to be nearly so prosperous, nor as attractive for immigrants. It will be quite a different world...