Roughly what would be the effect of global thermonuclear war on the planet over the course of 200 or 300 years?

I was watching PBS and it said that if a nuclear war broke out right now, and countries even 2 little ones, started throwing nukes at each other, the world would be in a nuclear winter, uninhabitable, within 28 minutes. If more nukes were used, the less time we would have. Just imagine, you see a flash, 28 minutes later, the end. The oceans would turn to battery acid, that's how high the pH level would go. The air would be deadlier than arsenic and cyanide. The temperature would never rise and storms would plague the Earth for about 60 - 100 years. Humans could survive they said, but they would have to have food and water for 100 years underground in a protected bunker. Nearly impossible to have that much to survive right? Hope this helps!!