The current maps, which largely correspond to prevailing winds too, indicate that departing from anywhere in western Europe south of Britain anyway, one winds up most reliably funneling into the Caribbean. Columbus could be unpredictably blown northward--just as he could have been OTL, but by and large one arrives in tropical latitudes. I would expect any consistent diversion northward relating to starting from a more northerly port (any likely French port anyway, certainly any south of the tip of Brittany) would at most be diverted to Florida, or perhaps the site of Charleston, but the most efficient and reliable landfalls would remain in the Antilles I would bet.
Given the differences between 16th century Spain versus France, it is entirely possible that the subsequent evolution of power, assuming it even stays in French hands, would be very different in the New World, but my bet would be that while it might be strikingly of a different pattern in detail, in overall thrust and impact it would parallel Spain's behavior pretty well. One big difference might be that the Papacy balks at granting the French monarchy a land grant of the entire Western Hemisphere--I believe the Popes were much more sympathetic to the rulers of Spain than France, given recent French adventures in trying to vassalize northern Italy and particularly the Popes via schemes to control Rome itself. But of course the OTL Treaty of Tordesillas was a largely scorned dead letter to rivals of Spanish and then Hapsburg power even before many of them broke ties with the Roman Papacy completely. France OTL remained Catholic, though large parts of its population becoming Huguenots came much closer than Spain ever did to a Protestant regime there too--still pretty far from it though is my impression.
I would also think that to match anything close to the pace of expansion the Spanish achieved in the Americas, many feats of French power in Europe itself would have to be stunted to provide for the diversion of manpower and resources westward--of course fairly soon the New World conquests would start sending back rich tribute, and perhaps the French monarchy would respond more creatively than the Spanish rulers to maximize returns to the imperial kingdom itself, taking a greater share of the overall enrichment of western Europe in the 16th century, and thus be in a position to sustain both near-OTL feats in Europe itself while also matching OTL Spanish rates of expansion in the far hemisphere.
In that case, or anyway in the case French power overall projects west on a scale close to what Spain sent OTL, I suppose a similar itinerary and timetable to Spain, though style might differ. First intensive focus on the Antilles, mainly the big islands of the Greater Antilles soon centering on Cuba, then a thrust across the Gulf of Mexico to the Mexican city-states and nations. The French might arrive somewhat earlier (but if so, in much smaller sustainable numbers) in Mexican and Mayan coasts, or be delayed somewhat, but sooner or later, within a couple decades give or take, they'd control central Mexico to the Pacific. I envision it as a somewhat more methodical, more centrally controlled, operation, with the French monarch sending trusted intendants and largely bypassing traditional feudal power with a monarch-centered gentry system. Vice versa I intuitively guess the French would be somewhat more liberal about the origin of the various conquistador-equivalents, being more reluctant than the Castilian regime in Spain was to send over either nobles or peasants of core French origin but far more willing to trust various diverse European foreigners to take allegiance and share in the spoils fairly systematically--they'd pick up a lot of Irish clients for instance assuming the British Isles develop on close to OTL lines. Or Italians from states allied to France, or a leavening even of individuals from nations that were France's worst enemies. If French power in Europe surges northward they might include a lot of Flemings, or Germans from the Rhineland.
Being nationally diverse would be offset by a strongly organized central French monarchial rule. Perhaps the French would also prove more liable to accept people of Native American or eventually even African slave ancestry as integral to the French-ruled regional elites--to be sure, the Spanish did try this, in the regions of the Andes taken from the Inca empire anyway, only to find the native chiefs leading a nativist uprising. If the French have similar experiences they might turn to a more stringently European-background ideology, but I believe OTL the Spanish reverses reinforced a deep tradition in Iberia favoring "pure blood," whereas a French regime might turn to that rule less reflexively and with more compromise--assuming that this ever paid off for the monarch. I think it would as often as it backfired anyway, and so the "tone" of a French Empire of the Indies might be different than the OTL Spanish one.
And its scope might be more limited. The winds and currents bring one in to the Antilles as noted, and so OTL the Spanish invested in controlling the entire sweep of the archipelagoes from Cuba to Trinidad, and thus tended to somewhat secure colonization of the entire sweep from Texas to Venezuela. If the French match this they preempt the route to Peru as well and probably secure the former Inca zone into their empire in the fullness of time.
But they might fail to hold the whole line of islands, and perhaps some rival European power can lodge themselves on the Caribbean coasts. Or of course divert northward to the North American mainland coast--but one reason attention went south OTL was that sailing around the southern tip of South America was the only westward passage into the Pacific; there was no usable northwest passage though of course a lot of effort went into looking for one.
Despite probably acquiring a network of allies in Europe, for reasons I've mentioned they might not be as well able to focus on American colonies and so perforce leave areas the Spanish moved to preempt open to invasion by rival European powers.
Certainly in the OTL, Portugal already had the claims on the trade route around southern Africa to the Indian Ocean, and infrastructure to back it up, and I think the evidence is good they had already stumbled upon and mapped parts of eastern Brazil including the mouth of the Amazon before 1492, but kept it secret as a diversion and attractive nuisance to their main circum-Africa venture. If France faces some tight limits on westward power projection they would do well to come to understandings with Portugal and stay out of their way--which means control of the Rio Del Plata region would be left by default to the Portuguese, until various powers seek westward passages and find the routes around Tierra Del Fuego anyway--at that point perhaps a policy of appeasing Portugal might pay off in France being favored to settle there too, or perhaps Portugal is too weak and tightly focused and lets some rival power (Spain, say) slip in and establish themselves. Whoever controls Argentina (of latter days OTL) probably has the inside track of settling Chile too, even if France already is established northward. Chile was chiefly valuable to Spain as a way station between the Atlantic and Pacific coasts, so if the French control the coasts from say OTL Peruvian shores northward to Baja California (Bas-Californe?) a rival holding Chile has less prospect than Spanish settlers there did OTL unless the French don't care which nation's merchant hulls appear to trade in their Pacific ports--which seems highly unlikely indeed! Perhaps the dynamic will be to drive French annexation of Chile but then neglect it because it is held mainly to preempt foreign bases to threaten the French New World Pacific coasts to the north. Of course as the arts of navigation improve the French might seek to communicate with these coasts via French ships running all the way from the Caribbean around the Horn without stopping anywhere on the Atlantic, then Chile would start to become far more trafficked.
Meanwhile, I also think that even with tighter bottlenecks of resources and manpower, the French would not neglect the northern coast of the Gulf of Mexico--OTL Spain did make some efforts there to be sure, but mainly just to establish mission outposts that did not expand very far. I think the French would start with the same bases, plus those they later established OTL in their Louisiana venture, but if they can hold on to the Caribbean islands, they won't find much interest in developing plantations on the north coasts--but they will gradually expand trade networks, associated with missionary activity of course, up the Gulf rivers--including of course the Mississippi itself. OTL Louisiana would be founded far earlier, and range from the OTL Texas to Florida peninsula Gulf coasts, but spread inland via Native American trade contacts pretty slowly, but steadily, making "ATL super-Louisiana" politically and culturally more like OTL New France--relatively few settler zones (though more scattered into the numerous river outlets, most heavily on the Mississippi of course) but vast tracts of land not only claimed France but with forms of French power lightly but widely spread on the ground, in the form of Native national alliances. French power might be near zero in the Great Lakes for instance, but at least they would have some reliable intelligence coming in from that distant zone steadily, and would be invited north by the Native associates in the French trading system if some other European power--England, or maybe Denmark, say--tried to invade.
On the Atlantic coast, they surely would secure Key West, quite possibly the site of OTL St Augustine, and maybe the site of Charleston. And perhaps early on move into the Chesapeake Bay but again I suspect on a missionary leading to trade relations basis, mainly to preemt other powers from setting up bases rather than expecting any value from the colonial outpost.
Do you know what the biggest factors of where he’d land would be? I assume it has mostly to do with where he departs from and how the ocean currents are during the voyage, given that Cartier and de Verrazzano landed in two completely different places.
