The hypothetical only makes sense if we postulate an identical planet, except the laws of physics are such that nuclear weapons are about as easy to build as gunpowder weapons. That is, you dig up some radium and mercury, mix in a little jeweler’s rouge, and set it off with the emanations of the eighth ray. And the whole thing goes up like Little Man and Fat Boy. You don’t have any delivery systems other than horse-drawn wagons or fire ships. You could have some sort of fuse, but setting one of these things off is going to be pretty close to a suicide mission. No ICBMs or long range bombers, no industrial revolution, nothing changes except that some alchemical formulation releases tremendous amounts of energy on the scale of nuclear weapons. And we’ll also say that this alchemical process should have the same sort of secondary effects as nuclear weapons, just to keep it simple–fallout, and contamination.
And we’ll say that the spread of this secret will pretty much follow the time scale of the spread of gunpowder.
As was pointed out, the use of alchemical weapons means you’re destroying the very thing you’re trying to steal. Blow up a city and you can’t conquer the city. What’s the point of that? Populations are low, productivity is low. Wealth is produced by land, but land requires workers who only produce a tiny surplus to skim off. The reason you want to conquer a land is so that you can rule over it, and the more productive the land and the people the more rich and powerful you are.
In the medieval and ancient world cities existed because they were defensible against attack. The enemy army moves up, everyone runs into the fortified city, and the enemy has to besiege the city. Except now that doesn’t happen anymore. The enemy just sets off an alchemical bomb and blasts the fortified city off the map. But now the city is gone, so what was the point?
So what results is a much more decentralized world. No fortified cities, just farms and villages. Nomads like the Mongols rule the world, since they can wander wherever they like and loot and pillage the countryside and destroy any fortifications and any concentrations of force. Everyone is much poorer since trade centers are smoking craters. Populations are much lower. Formerly productive lands are contaminated wastelands. The good news is that since most people were already subsistence farmers, they continue to be subsistence farmers, you don’t have the mass die-offs you’d have in the modern industrial world as trade systems break down.
People would survive, but at much lower population densities and wealth, and any concentrations of wealth and population get eradicated with almost no possibility of defense. Warfare would consist of rapidly moving raiders looking for enemies trying to deliver alchemical weapons, and rapidly moving raiders trying to deliver alchemical weapons. Fire ships mean naval warfare is obsolete as are coastal cities. All you have left is viking-style longboat raids.