Consider this: According to a character in New Vegas, 77 warheads targeted Vegas and the surrounding area. In the entire Mojave area, there are few targets of strategic worth (in approximate order of their strategic value):
Nellis AFB (Houses significant air power assets that could be used in a counter attack or any future military operation)
Hoover Damn (Provides power and flood control. Destruction of this asset would cause region wide blackouts and brownouts reducing the effectiveness of any activity in the region for a significant period of time)
Las Vegas (Home to ~2,000,000 people. The economic impact of the destruction of such a city is minimal as is the military impact as the economy of the city is based almost entirely around vice and tourism rather than meaningful production).
That's really it. Three targets. Three nukes is all it would take. If China allotted 77 to the region, then I suspect you get a sense of the scale of the war. Hell, DC should have been dust not rubble. The same would go for major industrial and economic centers. That means NYC, Detroit, Pittsburgh, LA, Seattle, Dallas, Houston, Atlanta - all of them would be dust.
This leads to a key problem with the fallout universe: a lack of logical consistency. If Vegas warranted 77 nukes, why didn't Pittsburgh, former home of the US steel industry (and, if the game is to be believed, still home in 2077 when the bombs fell) warrant at least a few?
The bottom line is simple enough: there are several thousand targets of strategic worth in the US, from ports to population centers, to industrial centers to military targets. The three targets in Vegas were more than 20 apiece. That implies that the exchange sent tens, if not hundreds of thousands of bombs to the US alone. You can bet the US responded in kind. Any nation not hit would suffer the perils of nuclear winter, extreme fallout, sudden power vacuum, and an utter lack of energy resources. The war was, after all, the result of the inevitable depletion of oil reserves. Before the great war, the Middle East destroyed itself in a local nuclear exchange.
So, to put it simply, even if a nation wasn't hit with a bomb, it certainly all but fell apart in the aftermath. The fallout lore more or less indicates the world of humanity all but ended inside of a few hours. Few who were not protected by a vault (or an equivalent) survived.