Alright then, I'll give it a go. This is going to be a long, rambling ride through my thoughts, so be prepared.
France in the late 1400s is in many ways unique in Europe. It is at the time one of the most populated areas in Europe--13 million in 1483 according to Wikipedia--and has just come off of the war that arguably created French national identity (the Hundred Years' War, if you're not familiar). On the face of it, 1400s France is a very successful state, with a well-developed military culture, a strong economy, and a powerful demographic weight. However, this belies the far more complex truth of the matter.
Historically, France has been one of the least centralized states in Europe--not something we often associate with France given the Ancien Régime, but it must be understood that said system emerged directly as a result of its medieval history. During the Norman Conquest, for instance, the crown was so weak that the great princes who controlled the various duchies often treated with foreign powers on their own and more or less ignored the Capets. While this had changed somewhat by the late 1400s, due to the centralization necessary for the Hundred Years' War, the nobility was still very strong and very influential. To compound this, the church controlled some 40% of French property (seriously) and often collaborated with the nobility against royal policies in order to preserve its traditional power.
At the same time, you have a peasantry that is greatly different from the idea of the feudal serf. While serfdom did exist at the time, it was far less strict than in other areas, meaning that peasants were not as tied to the land; indeed, social mobility often depended on geographic mobility, and many peasants rose to become the influential guildmasters and merchants that formed France's economic core precisely through moving and trading.
In a sense, you have two economic and social worlds existing in unspoken tension; the traditional, feudal, authoritarian world of the nobility and the church, and the burgeoning proto-capitalism of the townships, which often ruled themselves. Sitting atop both of these is a crown that, while more powerful than it has been in previous centuries, is still reliant heavily reliant on the nobility and the church to get anything done.
So. Let us then say that someone in French employ--perhaps even Columbus himself, if foisted off by the Spaniards--discovers the Americas in the 1490s (let's say 1495 to take into account delays due to the Spanish rejection). While it's possible that this explorer ends up in the same place as Columbus did IOTL, I find the odds to be vanishingly small and boringly convergent. So let us say instead that alt-Columbus lands on the northern part of the OTL Eastern Seaboard, somewhere a bit north of Manhattan in OTL Connecticut on the Long Island Sound. What, then, are the values of this new land to the French? No sugar, no spice, not all that much nice. Not a whole lot of gold to be sure.
But the-land-that-would-never-be-Connecticut holds great value in reality, due of course to fur. Fur was ridiculously plentiful in the early days of colonialism, and the Long Island Sound is a perfect place to profit off of it. Moreover, the timber found there is of great strategic value to a future French navy, and of course it is decent, though not great, land for tobacco cultivation when that is discovered.
I figure that most likely, the Crown uses this as an opportunity to gain advantage over its subjects and thus establish an alt-Ancien Régime about a hundred years ahead of schedule. Most likely a form of colonial company similar to OTL is formed, with hunters and trappers in service to the crown travelling to the New World (not sure on what it'd be called by Europeans in this ATL, perhaps
Colombie or something similar) and bringing back furs. Over time, it's likely that these fellows would make contact with native groups and tap into the continental trade networks (which likely collapse and then quickly reform as per OTL due to plague), gaining access to gold and tobacco from the south, and even possibly maple sugar/syrup from the north. It's likely that there are some settlements made here and there, but these I expect would develop largely as trade hubs rather than areas of heavy settlement and control. There's likely to be a military presence, but I would expect the French to be a LOT less heavy-handed than the Spanish. No conquistadors here most likely.
Let's assume, then, that France more or less has the Americas to itself for the first, oh, five decades or so, with other powers tentatively exploring and making contact (likely the British are among the first). The monarchy thus gains an economic and thus a military and political advantage over its enemies, and is most likely heavily invested in using this. What does the crown do with its new wealth? Well, first of all, the "beautiful 16th century" (in reality 1475 to 1630) is probably magnified even further. Wealth from the New World will inevitably trickle into the townships, likely granting them even more power early on, and an increased reliance on merchants and traders for the stability of the French state. The nobility likely weakens as a result; we may see widespread discontent in the form of civil war, which the nobility probably loses given new wealth. With the rise of townships in wealth, there is probably a corresponding rise in urban population, further weakening both the church and the nobility.
Ah, but it's foreign relations that everyone is interested in, no? In this arena, I figure a more successful Italian Wars on the part of the French, given, again, the influx of wealth. A French Milan is certainly possible, and perhaps even further (French Sicily, anyone?). Brittany is almost certainly incorporated earlier, given Nantes' immense value as a westward port. If indeed there is a Reformation, which there is likely to be given the longstanding institutional corruption in the Catholic Church, it likely goes worse for the Protestants, given that ITTL, France has no reason to side with them (by which I mean side with the German Protestants, certainly not the Huguenots) given its strengthened position, and may in fact be on the opposing side, which would mean French gains in the HRE. We may even see a reversal of the French + Protestant states vs. Catholic League dichotomy that emerged at certain points, with Spain using the Reformation as an opportunity to weaken France. Protestant on Catholic civil strife may occur as per IOTL, but I figure the Catholics win even harder than OTL as long as the Protestants don't actually convert the monarchy (which is more possible than you might think).
With that being said, none of this actually talks about the Americas, SO most likely in the mid-1500s, other powers begin to establish claims in Columbie, likely Britain and Spain first. Are there bush fights over this, as per IOTL between the French and Britain? Oh yeah. I can see burning trading posts, short and bitter wars which threaten to become larger conflicts, the usual. However, with the 'model' of colonialism being trade-based, I doubt we see many settler colonies. Even Spain, most likely, does not conquer the Aztecs or Tawantinsuyu, given that as far as they know, that's unfeasible without the model of Cortez. Keep in mind that conquistador colonialism evolved purely from the luck of Cortez. There are certainly settlements and communities in the New World; likely the Caribbean as per OTL becomes heavily populated by European settlements, given the low population of natives and the effects of plague, which will unfortunately probably wipe out Caribbean groups almost entirely, against as per OTL. But on the mainland? Most likely trade hubs and allied native nations, which is the more interesting point here.
With conflict emerging over trade, we (first of all) will see the development of a French navy, probably using NAmerican timber. We will also almost certainly see a patronage-based relationship with native nations, as various powers ally themselves with the Mohawk or what have you in order to use them essentially as auxiliaries against their enemies; in exchange for the normal trade goods, and of course, weapons. I figure European technology filters in slowly through these avenues, and we see native nations far away from the Europeans (and thus far away from the likely constant low-level conflict over trade) developing and consolidating against more technologically primitive and thus unlucky groups. I wouldn't be surprised if we see the emergence of a strong power in Mexico and among the Cherokee and similar groups in the southern *United States. Over time, it's likely that these powers will demand a more equitable relationship, which may in and of itself lead to war.
What would the future look like after this? I'm not sure. But these are my disconnected, rambling thoughts on the matter. Pass down your judgment with mercy, O Great Caesar.
