One: In this setting, China is strong enough to conquer South Korea, which is 9th most militarily powerful nation on Earth (Japan is 10th, in comparison), and backed up by United States. Not only that, it also conquered part of Russia, the 2nd most powerful nation on Earth. Now...let me get this straight. Here, China is strong enough to take on both America and Russia. And exactly...just exactly how on Earth did Japan managed to stay independent? China has a bad bone to pick with Japan (which actually applies to entirety of Asian countries), and it tried to conquer Japan multiple times. Japan had magic? If it was only Japan that had magic, it might make (only slightly) more sense, but it wasn't stated anywhere. Did Japan just endured through the Yamato Spirit or something?
Why and how exactly is Mongolia staying as an independent naton?
Also, the League of Arabian Nations? The Federation of India and Persia? Is the author so unknowledgeable on international politics that he never heard about decades long, violent religious conflicts, all those tensions, especially between India and Pakistan?
Why on Earth would EU be split in two? Is it because much of Eastern European nations used to be communist countries? If THAT's the reason, why not just have them assimilated with the Federal Republic of Russia?
United North American Continent? Why on Earth would US want Mexico? Seriously, what does it have to gain?
How exactly did magicians prevent thermonuclear war?
It's explained in Volume 8 that magicians formed the International Magic Federation, an international organization with the stated purpose to halt the use of nuclear weapons in the war. Magicians were permitted to intervene in border disputes, and, regardless of their nationalities or allegiances, would cease fighting each other when they observed signs of the use of nuclear weapons and cooperate to stop their deployment, whether it was their country using them or their ennemies'.
What? That doesn't make any sense. That essentially means that IMF is a group whoses stated purpose is armed intervention, ala [[Gundam00 Celestial Beings.]] Magicians are heavily involved with military and government, considering that they are, as unconventional as they may be, bunch of scientists, it is most likely that they are funded by government and military. In order for IMF's intervention to be powerful, the magicians would have to be enough political influence on their governments, but considering the magic wasn't discovered until 2000s, I seriously doubt that they would have established that much political influence on such a short notice of time. Also, I seriously doubt that any sensible and self-interested government would even let any kind of organization to grow that large.
We actually have an organization that does something very similar. It's called the UN. Whenever Human Rights are violated, the UN steps in to... take care of it. The Magicians are obligated to do the same thing, except with Nuclear Weapons. Whenever a nation uses one, its own magicians, as well as everyone else's step in to make them stop, by whatever means necessary. The IMF is the body that keeps track of and manages who would be required to do that should the time arise. Nobody wants to be nuked, just like nobody wants to have their Human Rights violated, and just like we have a system set up for the latter, there is one in place for the former. Also, keep in mind that magicians are individually dangerous to a ludicrous degree, so they can make good on any threats they care to cary out. It's not unreasonable for a young average battle trained mage to take on a tank and its support units, alone, and win. An average mage(I'm thinking around lower First Course to higher Second Course graduate). When you get a group of hundreds or thousands of them, its the next best thing to a walking natural disaster, and that's not even counting that there are dozens of them who are one-person armies, or in the extreme cases walking talking super-nukes in their own right.
The simple answer for the existence of the IMF is the sheer personal power of magicians forming their core group. If a plurality of strategic class magicians, reusable and highly accurate nukes in their own right declare its formation and founding, there is not a whole lot a government can do about it especially if there is an idealistic or opportunistic government already prepared to give them shelter. Many magicians probably soon joined for the patronage, protection, idealism or sheer pragmatism. In a world shorn of nukes, magicians are the ultimate strategic weapons after all. Any country unwilling to play ball might see the IMF withdraw its members, many of whom may be vital to their military. Think of them as a rather militant trade union with a few members who can glass cities. Furthermore the stated goal, that of preventing the use of nukes, probably met with some degree of public support since decades of Cold War literature painted such nightmarish pictures of the nuclear holocaust.