I see I misread the post. Third war, not third world war. My bad. Yeah. A third war between Britain and the US could have happened, but wouldn't likely have become a world war. However, I can also see the US becoming Great Britain's lapdog, too afraid to go its own way.
I can see the US fighting a war against New Spain. Its population would still be growing if it lost the war of 1812, and expansion into the Caribbean would be out of the picture due to the British navy. However, that would be a pretty hard war for the weakened US to fight. The US could win some territory, but at a big cost. The Emperor was really proud of his north, after all, and had the northeastern frontier heavily guarded.