Well, the first thing that would happen is mass chaos as states collapse. All states rely on some degree of having the only legitimate use of force. If every single person suddenly had the ability to build their own armies, the power of states would be broken within days. Sure, no one person would be able to overrun the planet, there would be too many people competing for local resources, but violence would break out damn near everywhere as people explored, experimented, got into fights, and those fights escalated to military grade hardware or beyond. A small number of people would leave Earth to establish themselves across the solar system, and would immediately be pursued by tens of thousands more trying to copy them, prevent those people from building up, or just escape the violence that is occurring on earth. When anyone with control over a decent sized swath of territory can become a nuclear power within an hour, the planet would become a nuclear wasteland in short order. Someone would launch the first nuke, and after that it would become a free-for-all.
After the initial die-off and jockeying for power, the solar system would settle down into a new "world order" where people with the greatest amount of raw resources (and the numbers to defend them) would hold power. Those people would craft new societies that would take into account the fact that every citizen/subject/cult member/neighbor could be a nuclear power within an hour if they go unmonitored. I expect that most people would be under constant observation, or would be banned from acquiring the resources nessesary for mass construction. There would still be periodic violence as interpersonal arguments could rapidly escalate, but the existing leaders would probably implement policies designed to mandate escalation. Still though, the solar system would still probably be too unstable and dangerous for a lot of people long term, and so you would see a lot of commanders leaving for other solar systems, probably in groups of a few to a few hundred at most. These groups would gradually form new societies with new social norms that permitted survival in such a heavily armed society. Those that failed to do so would die in nuclear fire.
The good news is, even on a galactic scale there will never be a runaway force capable of overrunning the galaxy. Everyone would have similar levels of construction capabilities, and would be able to defend themselves even against enemies that threw the resources of entire solar systems against them. In the end, most solar systems would become vast fortresses dedicated to holding off inconsistent raids and attacks by people looking to acquire more resources (and ideally, loyal followers).