|The only chart I could find|
The world, right up until the invention of the nuclear bomb, was dominated by forces of consolidation & empire. A simple dynamic existed in that world: the larger your nation the more threatening it is to its neighbors. The tendency was to grow as big as you could, as fast as possible. Tribes were conquered to make kingdoms. Kingdoms were conquered to make empires. A lot of nations that we wouldn't now call empires were called just that back then: the Ethiopian Empire, the Holy Roman Empire, the German Empire, etc. Everyone got an empire.
Imperial colonization meant that the number of countries in the world steadily shrank. If the atomic bomb had never been invented it is quite conceivable that we would now be living under a one-world government of some type, and that a third world war would have produced it. Global communism maybe?
Einstein sent a letter to FDR about the possibility of building a nuke in 1939. The US then dropped 2 of the things on Japan in 1945.
After the bomb ,the force of consolidation was replaced by the force of fission. Endless proxy wars helped to divide the empires that formally existed, though the fissional process may have stalled a little with the end of the Soviet Union. Will China restart the process? All it takes is a new Cold War.
Given enough time proxy war may turn the whole world into patchwork of city states. Only way to stop this is to end proxy war by giving every nation the bomb. All nuclear armed states will work to prevent that of course, and thus, they will guarantee that the process produces the maximum number of countries possible.
The world becomes a patchwork. Everyone gets the bomb. War becomes impossible. Nations switch to financially manipulating each other's internal politics. "Diplomacy markets" take over. Final stage of human history = world ancap patchwork.
Then maybe one of the city states does full anarcho capitalism. Maybe it out-competes the regular patch. So the final stage of human history becomes world ancap.
Terrorism might actually be putting a break on the whole thing by delegitimizing secession movements. A genomics arms race could speed it up. As the number of countries with nukes increases the number of ways to have a cold war increases exponentially. Cold Wars led to colonization/expansion in at least two eras of human history: the European conquests and the Space Race. Perhaps North Korea adopts state capitalism and enters the space race. The more parties that participate in a Cold War the more irreversible it becomes. That is why the Moon Landing went no further while the European colonizers conquered the whole world. A two party Cold War ends when one of the parties withdraws, (The USSR), while a multi-party Cold War can only end if everyone withdraws, joins into a union, (like the EU), or gets conquered, (like the Chinese wars of unification).
All of this brings up another point: the gap between getting a bomb and getting a launch delivery vehicle is extremely dangerous. If you have a nuke but no way to delivery it then you have an incentive for someone to invade you. If you don't have a delivery vehicle, like an ICBM, you have little counter-incentive for them to leave you alone. But this creates an uncertainty, because you might be able to pull off a nuclear counter-attack against an invader. You want to have the launch vehicle figured out and the nuke procured on the same day, that way you go from being a subject power to a world power instantly.
Uncertainty is the basis of war. More uncertainty of outcome = more probability of war.
The intelligent strategy would be to make ICBMs available to every country, but nukes incredibly scarce and expensive. This preserves the power of large states while limiting their ability to get themselves destroyed. They won't do it of course, but they should. And no, all the world's nations should not be given nukes at once as that would throw everything into crisis. The collapse of an empire is always met with an increase in global conflict. Sometimes this even leads to world war. The transition to patchwork is best managed as a slow boil that disintegrates everyone's geopolitical power gradually.
Patchwork may be more of a prediction than a normative formula.