I think the best way to prevent World War 1 & 2 from firing is to prevent the Prussian unification and formation of Germany. No ascendant Prussian militarism humiliating France and you don't get the First World War (and thus no Second). The best way to do this is to prevent Napoleon from coming to power and co-opting the French Revolution. Without his ambition and military ability Revolutionary France either finds its footing and forms a workable Republic or the European Monarchies finish them off and re-install the House of Bourbon. Without Napoleon dismantling the HRE Prussia won't be able to consolidate it's hold over the other German states. Without Napoleon rampaging through Europe the HRE continues along with Austria in the driver's seat. Without Napoleon spreading his law code and the ideas of the Revolution this will be a very different, much more aristocratic world.
Another butterfly of this ATL is the Spanish and Portuguese Colonial Empires don't collapse as fast. OTL Napoleon deposed the King of Spain and put his brother on the throne. Then he took Louisiana back from the Spanish and sold it to the USA. The lack of legitimacy caused revolutions throughout the Spanish Empire and they lost a lot of their colonies. Regardless of all this the USA still will see the Port of New Orleans as strategically and economically important. Jefferson will probably try to buy it from the Spanish just as he did the French - I have no idea how successful that would be. Would the Spanish recognize that they can't hold on to their North American possessions in the face of American expansion? Or will they try to hold on to them? Historically even before Napoleon Spain is starting to decline and have financial problems.
Without the Napoleonic Wars Britain doesn't face a manpower crisis for the RN and thus doesn't go around impressing American sailors, so no War of 1812. Instead there could be a war between the USA and Spain. British policy at the time was to prevent territorial expansion of the USA by supporting various Indian tribes and confederacies such as Tecumseh. ITTL we could have a Spanish-American War of 1812, with the Spanish receiving support from British-backed Indian tribes. Provided that Britain doesn't get directly involved I think the US will win against Spain. The logistics and distance all favor the US. Spain will have the same problems as Britain did - they can't afford to concentrate all their forces against the US as the have a vast Empire to keep under their control and the cracks are starting to show. All it will take is a few US frigates managing to sink or capture a few treasure ships and suddenly the Spanish are in trouble. Another butterfly - with an actual hostile European power on its border and possibly blocking its expansion there will be more pressure in the US to have a professional standing military. OTL after the War of 1812 the US Congress realized the country needed a much bigger Navy and increased funding. The end of a Spanish-American War of 1812 could still see the US and Spain sharing a border and require a larger standing army. There's no telling what butterflies this will have for the various Indian Wars, Texas, or the Civil War. By the end of the 19th Century though I think we will see a United States that largely looks like OTL. There is just no one else in North America that can compete with the economic, demographic, and resource-rich USA.
So how does this effect the late 19th/early 20th Century? The two great belligerent powers in Europe in mid-19th Century were Britain and Russia, and I think this will continue in to the OTL WWI/II period. I think you will see a series of Crimean War type conflicts - efforts by Britain to curtail Russian expansion and influence, ie, the Great Game. The big question will be, will Russia fall to revolution as OTL? The good news is that I don't think the British would be dumb enough to secretly back the Bolsheviks as the Germans did during WWI in OTL, so Russia probably becomes some kind of Federal Republic or Constitutional Monarchy.
