If the American colonies had not won at Yorktown, they would have continued fighting and eventually the British would have given up because they were too busy fighting elsewhere.
The Battle of New Orleans was meaningless as the war had already ended with the Treaty of Ghent.
The North was winning the Civil War. Appomattox was the last battle, but if the South had won the North would have likely won soon after.
If the Brits had captured New Orleans, they would have tossed the unratified Treaty of Ghent.