As always when looking in this general "direction", the answer is simple: curb revolutionary radicalism, so that moderate Enlightenment remains utterly dominant. This will then avoid mass slaughter by frenzied radicals, while a lot of aristocrats and upper middle class types stay wholly in favour of Enlightenment reforms. Free trade becomes the norm in France (which becomes a consitutional monarchy after the fical situation becomes too critical to let things stand as they have until then), and an early Anglo-French detente commences. No Anglo-French wars involving republics or Bonapartes here!
This means no Quasi-War or War of 1812, either. In the USA, the more enthousiastic democratic reformers never adopt the more bloodlust-inspired French radicalism, and on the flip-side never get associated with the Terror in France. A youthful idealism of American virtues lasts longer, and politics are less acrimonious. Without the Anglo-French conflict, all sides agree that nothing stands in the way of American friendship towards both. An Era of Good Feelings dawns earlier. The USA, the British Empire and the French Empire slowly move towards becoming (at least de facto) a giant (more-or-less-)free trade zone.
Other nations are dragged into this orbit. Never invaded by France, and having an example of reforming successes rather than wild revolutions, the smaller countries of Europe begin to imitate the policies of the nascent "Atlantic alliance". Without the enlightenment gaining a bad rep in certain quarters (which was due to revolutionary radicalism and bloodshed), the enlightenment internationalism of the Enlightenment never wholly dies out. The old idea of an intellectual "Republic of Letters" that transcends borders slowly evolves into the ideal of an alliance of liberal powers, who do not wage war against each other, and attempt to hold themselves -- and the world -- to certain standards. this idea will fall short time and again in practice, as is always the case, but the idea lives, and remains a goal.
The Duch Republic eventually sees the intellectual heirs of the Patriotic Movement achieve their general goals. Reforms in the Habsburg Monarchy -- brutally cut off when the French Revolution escalated in OTL -- continue on. There are pitfalls, but no fatal falls. Through trial and error, things get better over time. The German states and the Italian ones are far less pressured to unite, as they are not invaded by French armies. They form confederal leages, mostly to ensure free trade, and likewise engage in free trade with the greater liberal powers. Germany in particular, never develops its anti-Napoleinic and thus anti-French and anti-Enlightenment type of Romanticism, and the HRE gets slowly transfotmed into a confederal Germany that the Frankfurt Parliament would have been proud of.
A Greek revolt still happens, and the joint intervention of France, Britain and Russia sees it succeed to a far greater degree than in OTL (liberating all greek-majority areas in the Ottoman Empire). This sees the Balkan states find independence as well. Divided on what to do there, the Austrians and Russians agree on making all Balkan states joint protectorates with guaranteed neutrality, with Prussia, Britain and France as observers to guarantee compliance.
Meanwhile, the loss of all its European possessions -- including Constantinople -- prompts the Ottomans to reform their Empire, bringing in countless Western advisors, and essentially "pulling a Meiji". The various ethnic groups etc. get guaranteed liberties defined in a charter of union, and the "United Ottoman States" are born. When ethnic tensions later flare up in the Habsburg monarchy, the Habsburgs actually use the UOS as an example, and form the multi-ethnic United States of Austria. Somewhat later, around the turn of the century, Britain and France begin to federalise their empires.
Meanwhile, domestic reactionary backlash is avoided most everywhere, since a policy of gradual reforms becomes the norm. Socialism never even takes off as anything more than a fringe movement, and improvement of workers' conditions etc. is managed through a succession of incremental reforms (none of which encounter meaningful pushback). Revolutionary sentiments remain largely absent altogether, and a mindset of progress and prosperity defines the culture of all "modern" nations.
Latin America, never thrown this way and that in the chaos of the Revolutionary and Napoleonic Wars, stay with Spain and Portugal a bit longer, and gain independence in a largely orderly way. Their politics are more stable as a result, and treaties of alliance with the USA and other liberal powers are soon drawn up. Spain and Portugal, meanwhile, enter a period of slow decline. Eventually, popular discontent ends the absolute monarchies, and the new, liberal regimes join the Atlantic Alliance.
By the 1920s, this alliance is formalised by a binding treaty, and includes such conditions as arbitrage amongst the members, lasting peace and amity among the members, free trade, and a dedication to a charter of universal rights. Members include: the British Federal Empire, the French Federal Empire, the United States of America, most of Europe (including the German Confederation, the Italian Confederation, the United States of Austria, the Scandinavian states, the Dutch Republic, spain, Portugal, the Balkan states and Greece), as well as all of Latin America and the United Ottoman States.
By the mid-20th century, the federal empires of Britain and France devolve greatly, with a lot of African countries and -- most noticeably -- India spinning off, but typically joining the Alliance as new members. Colonialism is eventually ended in a mostly responsible transition, which tries to ensure that all formerly colonised peoples get a chance to form their own nation-states on principles of their own sovereignty. At this same time, Russia -- an observer earlier -- joins the Alliance after holding referenda in Ukraine, Poland, the Baltics and Finland over independence. (The vote was universally 'yes', but since those others are also set to join the Alliance, there will soon be free trade again anyway.)
By 1980, the Alliance has developed into a military partnership as well, with most nations mandating tasks of any peacekeeping missions off to an international Alliance command. There have been no major wars between any members, and small skirmishes -- mostly involving former colonies -- have been dealt with astutely, for the most part. The Alliance, its capital in Zürich these days, is considering the membership applications of China, Japan, Korea and Persia. Already before any such admissions take place -- which they no doubt will -- the Alliance is the undisputed hegemon of the world. Many are tentatively beginning to talk about federalising the Alliance further. As a first step, an Alliance fund has been set up to help infrastructural development in the least developed member states and/or their regions, to get all the Alliance on a common level of development. It is the general stance that a universal currency cannot be introduced until that equalisation has been carried out. (To push for monetary union when members aren't equally developed, after all, would be insane.)
The world is largely at peace under the Pax Atlantica. In the distance, the United States of Humanity await: the promise of a global union, a planetary federation, that will one day serve (as many alreadty hope and dream) to carry our descendants into the stars. Does the Alliance banner not already stand proudly on the surface of the Moon? The future is bright.