It would almost certainly be proxy wars between the U.S. and China. With MAD in effect, that's pretty much what you've got. If the two countries ever did fight directly, however, it would be because of resource wars: a mad scramble to secure and gobble up what's left of the world's remaining limited resources, mostly fossil fuels. That of course brings us back to MAD, and there's really only one way this would ultimately play out if the two leviathan countries went to war 'face to face'. If you've ever played the Fallout games, basically that.
I don't think the Fallout lore's scenario is the only one that could play out. Even in Fallout itself, if nuclear technology that was replacing fossil fuels had evolved just a little faster the entire Great War wouldn't have happened.
I agree that proxy wars would be way more likely. If either side hit each other’s mainland it would seriously escalate the situation. I would think the largest factor in such a war would just be the overwhelming air and sea domination of the USA. If things escalated so the point where they had legitigmately declared war on each other the USAs first course of action would be to cripple Chinas capabilities. We’d hunt down and sink their aircraft carriers then blockade the whole country. China would have no choice but to hit as fast and as hard as they can because once the USA has domination of the air, it’ll be hard as hell to claw back. The USA can reach anywhere in the world, China’s capabilities are limited to the continent of Asia. They have some ICBM capabilities but not enough to be an existential threat. Japan and the other nearby allies would have no hoice but to fight, China would likely try to annex them quickly and deny US the area. The nazi ground forces could easily have crushed them but they had just enough fighter planes and the overwhelming Royal Navy in between Hitlers troops and their shore. Japan would likely find itself in this position. Russia would be the wildcard. They may help their Chinese friends or they may stay neutral and hold onto their economic ties to Europe, which would likely unilaterally aide with the USA. From there, you’d probably have bitter fighting in China itself. Missiles will be launched to allied bases but China would largely on defense. The USA would eventually have to land and face chinas land army. Both sides would have rediculously high casualties as China will be well dug in and the USA would still dominate the skies. I may be a pessimist, but I think that if two nuclear capable countries end up in a real existential brawl, we’ll end up using them. I think countries near defeat, the danger levels go through the roof. In war, all options are on the table, and circumstances dwindles them away until only the most extreme options are left. The whole point of a nuke is the deterence that if attacked, they’ll use it. In order for MAD to work at all, both sides have to be willing to use their weapons.
The Chinese wouldn't go to war with the US. They would just start spending their dollars and collapse the American economy.
Wars since the Second World War have been primarily war by proxy, heavily unbalanced, or between small powers. A conventional war of this nature would be Pandora’s Box — World War 3. The results would be as unpredictable as the first two world wars. No one had any idea what those wars would look like until they happened. I think you have a lot of freedom here, but my suggestion would be for things to go disastrously wrong and nothing work as expected.
Although doing that would also tank the Chinese economy since they have a massive trade imbalance with the US... if the Us stops buying Chinese goods they'll be stony broke
