The good news is that there was no useless War of 1812. The bad news is that there was no Louisiana Purchase, either; Napoleon was hardly going to sell large parts of North America to his British adversaries. In fact, it is virtually certain that the French would have opened up a new front against the British in America. How would that have ended?
It is unlikely that Americans would have been drafted for service in Europe, but it is easy to imagine a scenario in which French forces based in New Orleans would have been fighting Americans for control of the Mississippi. In other words, there probably would have been a Battle of New Orleans; it just would have been against the French, not the British.
In the end, superior American and British resources in North America would have prevailed, and the area included in the Louisiana Purchase would have become a new British colony. It would have been a difficult battle, however.