Based (a bit loosely [1]) on H.R. Percy's "Letter From America", in the Sandra Ley-edited "Beyond Time" collection.
The French got into American colonization more energetically early on than OTL, (Louis XIV had one of his manic moods) and had a big enough population base by the mid-18th century to win the equivalent of the French and Indian wars: later, they absorbed the British colonies. French-Anglo conflict was long a serious internal problem, with separatist terrorism taking place as late as the 1970s, but rapid economic growth over the last four decades and some political compromises have greatly lessened tensions by 2010 (althogh visitors from the French-majority states grumble a great deal about everything being bilingual in their states while the "Onglays" can't be bothered to learn good French).
Although English-speakers only are a majority in six Eastern provinces, they are scattered far and wide: overall, they make up about 25% of the population of New France. Native Americans have generally done better than OTL, although the government has long tried to make good French-speaking Catholics of them: the black population is also somewhat better assimilated than in our US (slavery came to an end in the 1870s).
The Republic was the leader of a coalition of nations in a cold-war type situation versus the Collectivist Union, similar to Soviet Communism but bigger on anti-nationalism and Maoist popular mobilization. The Union has recently collapsed, rather more messily than OTLs Soviet Union, and political fragmentation and civil wars are still ongoing. Alaska is a Russian "Taiwan", a constituional monarchy rather more populous than OTLs state.
Important allies include China, a 40% Christian (more or less: there are some odd local offshoots) republic - a working democracy, but a protectionist, import-substituting, bureaucratic one whose "lite socialism" has left the country, like OTLs India, relatively poor. And then there is the empire of Japan, "bastards, but our bastards" while the conflict with the Collectivists was going on, but now increasingly a *fascistic embarrassment. The Republic of Great Britain has a relationship with New France as prickly as OTLs France has with the US, but is still an ally and militarily formidable. France is nowadays something a satellite of its overgrown offspring.
Central Europe is still scarred by the Short War of '77, which involved both chemical and radiological weapons. Post-war, the Allied Powers created the "MittelEuropa Recovery Zone" as a framework for the economic rebuilding of the region, which been renamed the "MittelEuropa Development Zone" as the local economy has picked up: the expansion of the union to the rest of western Europe has been often talked about, but the current free-trade regime seems sufficient to most without adding in a meddlesome Prague bureocracy.
Latin America is more closely economically tied to New France than it is with the US OTL - free-trade arrangements with fellow Catholics and Latins came into effect fairly early on. The Spanish Empire's efforts to devolve control in Latin America by creating sub-kingdoms within a overall Spanish Empire were only partly successful, with much of Latin America eventually going republican (or beyond, in the case of Gran Columbia, which is an anarcho-socialist republic).
Africa is largely post-colonial, only the Italians holding stubbornly onto their mineral-rich holdings. It is broken up into somewhat fewer and larger chunks than OTL, some pro-democracy (at least in theory), some still Collectivist, more English and Dutch and less French than OTL (French colonization efforts were confined here to North Africa and the western Sahel/Senegal area). South Africa is a loose federation of African and Dutch (well, at least Dutch-speaking-plurality) states. The continent is poor and messed up, but not as relatively badly off as OTL (cynics might argue that is because the locals haven't been independent as long as OTL).
The Arab Federation is more or less a democracy, although oddly the same party keeps winning the elections, although at least the Presidents don't overstay their terms. The south Indian republic is a prosperous democracy, and formed a keystone in the Indian Ocean portion of the democratic alliance's military plans: it still hosts a lot of British and New French troops, since the North Indian situation is to say the least fluid.
Democratic leftism is more anarchist, anti-state than OTL, and has even less patience with the Collectivists than our Social Democrats had with the Communists. Technology is a bit behind our TL, varying from late 1960s to mid-80s depending on the field: the first satellite wasn't put in orbit until 1983, and nobody has been to the moon.
Bruce
[1] Although we do have an unreliable narrator, there was a definite smell of "French America is greatly inferior to British America! The Commies are gonna do much, much better" about the story which I found a bit annoying.