Fairly random: a map based on "Assault on Fat Montain" by R. A. Lafferty, in Sandra Ley's "Beyond Time" collection. The story is humorous and somewhat fantastical (Lafferty. Duh.), and aside from the existence of France, the UK and Mexico, we are quite uninformed about the world outside of the US and the Free State of Appalachia, so I just made up stuff.
In this world, the mountain men, free-soilers and independent-minded homesteaders of the Appalachians and points west rebelled at being run by a Federalist Washington-based government dominated by southern planters and east coast urbanites, and the US west broke away to form their own anarchic republic. As of modern times, under the banner of a fat pig with the light of heaven shining down, Appalachia prospers. Appalachia is rich, democratic, a bit anarcho-capitalistic but with a powerful populist political wing, heavily armed but uninterested in foreign ventures, with not too many laws but lots of amateur lawyers and a very definite set of unwritten "rules." It is the world's most productive agricultural nation, and a bustling manufacturing nation to boot, but it has been falling behind the Prussians and the Japanese in high-tech of late (Appalachia has a lot of schools, but doesn't have a massive government-driven R&D program). It is welcoming of immigrants and nowadays racially tolerant (although it doesn't have too many African-americans: free-soil always meant keep out the black people) and the native American population is more socially integrated than in our world.
Relations are prickly with the US, which has tied once and lost twice in wars with Appalachia. The last lost conflict, combined with economic troubles deriving from a loss of industrial competetiveness in the 60s and 70s, led to the establishment of the current "Communalist" (this timeline's equivalent of Socialism) government. The US also has few blacks, having shipped most of them off to Haiti or the two "Liberias" in Africa after slavery ceased to be profitable. US Communalists do not get along with the "Latin Block" European states: ideologically, the International Communalist movement does not have so much as a programme as an ongoing pie-fight.
US filibusters created the "Red River" state on the borders of Mexican territory, but it failed to flourish, and after joining the US in the last war with Appalachia and losing a third of its territory to Appalachian ally Mexico, is now thinking of joining the Free State. (The Central American Empire of Walker VI is another story altogether).
China, formerly carved up into colonial posesssions, is now down to 3 pieces from seven. So far, attempts to finish the job of reunification through negotiation have fallen through, and a new war seems to be in the offing. Russia, after a lost war with Prussia and the UK, a failed revolution and a second half-successful one, starts the 21st century under the theocratic Rodina party, and is generally isolationist and hostile to a "corrupt" outside world, although it does produce a lot of Orthodox missionaries. It is also rather more populous than OTL, having missed out on the worst of our 20th century and its conservative citizens having only pretty recently ceased having large families.
One reason Russia has missed the horrors of our timeline is that the 20th century has generally been more peaceful: a sort of Japanese-British-Prussian Entente Cordiale has worked together to slap down threats and maintain something in the way of international order. (Japan is a long-time British ally, with a somewhat Anglicized and twee upper class culture, while the British and Prussian royal families are closely tied by marriage). Prussia is nowadays a pretty tolerant society, with Poles and Germans on an even footing (alas, both warmly agree that the Jews are probably Up To Something). The three powers keep a cautious eye on the Russians and the Communalist states and try to nudge the Appalachians out of their traditional heavily-armed neutrality.
Africa has been decolonized, and their is much economic socialist bungling going on. The Unificationist Movement has so far only taken power in two African countries, but calls vigorously for the unification of all sub-Saharan Africa, by hook or by crook. The Republic of New Israel does not get along at all well with the East African Federation, but African armies are even crappier than Arab ones and the locals were thin enough on the ground at the time of settlement that the Israelis do not have much of a "Palestinian" problem.
Spain is the most extreme of the Communalist powers, and took advantage of a popular rebellion in Portugal to invade and establish a puppet government. Other powers snapped up the Portuguese colonial empire, with the Brazilian military Junta grabbing Angola. A Portuguese government-in-exile still exists in the Azores under British protection. The Ottoman Empire survives as a loose federation which has actually expanded in Africa and Arabia through association, but is now being challenged for leadership of the Islamic world by an autocratic modernizing regime in the Mahgreb. (Think Saddam's Iraq, but with less megalomania).
The Kingdom of India is somewhat shakily unified under Charles III of India, whose marriage to a Hindu princess helped smooth things over with a populace who suspected his loyalties. (Nowadays, there is some pressure on the Heir Apparent to marry a Muslim, to balance things out between India's squabbling communities). Canada and New Zealand are more populous than in our timeline, and more closely tied to Britain in a federal union: although Australians are still nominal subjects of the King, they have refused to join the closer union and have gone their own way under a populist movement looking to Appalachia for inspiration.
Technology is a bit behind ours in some ways (no ICBMS and fairly poor consumer electronics), comparable in others (lots of nukes floating around) and ahead in some ways - most particularly genetic engineering, where the Appalachians lead the world in developing better crops and superior farm animals. (It is rumored that the French and the US are using genetic engineering for less pleasant purposes, involving germs and bioweapons).