It wasn't just China although I believe there is more than enough to tells us they were the ones who ultimately made the call to end it all with nukes.
Their actions would have dragged in the Soviet Union. Although China had replaced the USSR as America's number one threat, the USSR was still around and would have been an ally of China. Fallout 4 has a mention of the United States running scenarios in which both the USSR and China attacked the United States.
Not much is known about whether or not NATO and the Warsaw Pact were still around by the time of 2077, but if they were that would be the alliance that dragged the world into a World War.
I would make the case that NATO was still around, even after the fall of the European Union and United Nations. At least the bare bones of it. I say that because I can't imagine that the United States would sit back and let the Communists take over all of Europe. So I would figure that the USSR and the United States still had an understanding that Europe would be left alone. They can fight one another all they want but the United States and the USSR would not get involved.
America and the USSR had diplomatic connections and they both understood Mutually Assured Destruction so they would avoid it. China was the wild card. Under Mao the Chinese Government believed they could win a nuclear war. Something along the lines of "thousands might survive in the United States and the USSR but millions will survive in China." That idea scared Stalin/Khrushchev. That attitude to nuclear war would have lasted till 2077 in the Chinese leadership.
The USSR wouldn't think to invade America but China did. So one would have to figure any intelligence agency worth its salt would rank China a more serious threat. But that being said it would be hard to imagine China invading Alaska without the USSR's help. The Sino-American War could have been yet another proxy war between America and the USSR.
Anyways, so when China gives up and fires off it's nukes. The United States has to fire theirs. This triggers the Soviet Union to launch theirs. What's left of NATO uses theirs. And by then anyone who has them uses theirs. It was a World War. This is just an exercise in how it could have played out.
Edit: We also know that even though Japan had actually attacked America in World War 2. America still saw the Nazi's as the greater threat. Europe couldn't be allowed to fall to Hitler. I could easily see that happening in the Fallout Universe even after China invaded Alaska in 2066. America would still protect Europe from the USSR with threats of Nuclear Annihilation.