https://en.wikipedia.org/wiki/Mount_Weather_Emergency_Operat...
https://en.wikipedia.org/wiki/Raven_Rock_Mountain_Complex
https://en.wikipedia.org/wiki/Project_Greek_Island
https://en.wikipedia.org/wiki/Cheyenne_Mountain_Complex
With the rise of solid fuel ICBM and then MIRV leading to the truly massive number of warheads pointed at the US, the US switched to airplanes for the most important continuity of government issues, figuring that the skies 30,000 above the US will largely be secure (presuming the plane is appropriately EMP shielded) due to the many US geographic advantages, and so it is the best place to ride out the initial attack and then take stock, get to somewhere safe, and figure out what to do from there.
https://en.wikipedia.org/wiki/Operation_Looking_Glass
https://en.wikipedia.org/wiki/TACAMO
https://en.wikipedia.org/wiki/Boeing_E-6_Mercury
But the North Koreans can have no illusion that the skies above their country will be safe: there are several major enemy airbases a few minutes from their border, their entire airspace is routinely surveilled and powers hostile to them have made large investments in stealthy air superiority fighters, so the air is not a safe place for the DPRK continuity of government plans. The DPRK does have trains but I would not consider those safe in the event of a major war, since rails are difficult to keep secret.
https://en.wikipedia.org/wiki/Taeyangho_armoured_train
So bunkers are the best they can do, given their circumstances.