The Shi washed up in America after the War. Chinese American's I am sure were under alot of scrutiny in the years leading up before the Great War. I wouldn't be shocked to learn laws we passed so there couldn't be large groups of them at any given time, which would go against freedom of association and all that but it when we look at America's past and how things went down in the Fallout Universe... it might just have happened.
Another possible thing that could have happened was that survivors turned on Chinese Americans and so on, to vent their rage at what had happened. An escape goat.
As for Chinese Bombshelters. Their leaders and VIPs I am sure had them, big fancy ones. Civilians and cities and towns might have built their own shelters as they saw fit. But the over all mentality was that their population would be so large that they couldn't all be nuked. So an over all national protection program might never have been put into place. Heck America didn't have such a program, at least it didn't end up that way with the Vaults.
China would have been forced to use their nukes as a last ditch "screw you all" action. To play the spoiler. They were being overrun by America, they saw no way of winning so they just flipped up the board. They launch, America launches and then everyone does. "Game Over Man! Game Over!" I think China would have been so over confident with their invasion of Alaska that they wouldn't have had a Plan B. As in they thought they would win. "How could we not win right?" type thinking
Maybe in the years of 2066 to 2077 there was some attempt to put together a Plan B. But I think what resources they would have had would have been put into Plan A which is "Crush America and Win!"