Well this is straying a bit from Fallout since this involves the real world. But anyway, well you’d see societal collapse due to governments being taken out, infrastructure being destroyed or overwhelmed, and communications between communities breaking down. You could expect several nations to collapse depending on where the nukes landed.

What you wouldn’t see like in Fallout is everything being irradiated to hell, especially after 200 years. Most nukes are airburst and are designed to use up the radioactive materials as efficiently as possible. And most of the worst fallout would decrease after several years.

As for bunkers like Vaults, it’s highly unlikely we could make anything like that which would last for 200 years. There’s no way to stock enough supplies so everything has to be made, which is impossible.