The French Client Republics would not be annexed into France to avoid war, and would simply remain as puppet states to try to influence other nations in Europe (as well as worldwide) to become republican.
Alliances immediately after the war would become the Anglo-Prussian Alliance, the Austro-Russian Alliance and the Franco-Ottoman Alliance.
Without concurrent Coalition wars, there would be no need for British interference with the United States, and thus the War of 1812 does not exist. As such, tensions between the British Canadians and Americans would rise severely, as American presidents aim to "liberate" the Canadians from the hands of the British, who they feel are oppressing them (which would not be tested during the War of 1812).
Without Napoleon to invade Spain, the South American nations do not have a reason to revolt. Liberal ideals in the continent are hard to come by, and relative peace in South America allows Spain and Portugal to keep their colonial possessions until the mid-1800s.
During the Lower Canada Rebellion, France decides to join to help free the Lower Canadians from British reign, resulting in their independence as Quebec.
With the US not getting Louisiana until much later, the British have a larger time span to control New Caledonia and avoid a major boundary dispute.
With the bloody war that gave Louisiana to the United States, Mexico begins discouraging immigration from the United States, who fear more land will be taken. As a result, Texas stays Mexican and no Mexican–American War occurs.
With the changing alliances in Europe, the United Kingdom and Russia enjoy better terms with each other. As such, the Russian government does not consider the British a threat in annexing their colony, and keep possession of it until World War I.