The radiation would be long gone, nature would have taken over and everywhere would be green and society would have been rebuilt about 150 years ago.
agreed, after 200 years, there would be more forests than wastelands. take Chernobyl disaster for example. it released as much as 400 times the radioactive contamination of the atomic bombings of hiroshima and nagasaki. after 24 years, the trees and some animal spices are reported to have taken over the area, even where the explosion occured.
as for the cars, there would probably be hundreds of highly trained mechanics and engineers living in vaults. according to fallout bible, during the great war, there were 122 vaults and 1000 residents in each. even if only 1% of these people were mechanics or engineers (they would have a fatal importance for keeping the things go) and even if half of these vaults were destroyed in war, that still makes 610 well trained engineers in the wastelands. they wouldn't have to try to understand how to make vehicles because they would probably have schematics for every engine that would be necessary after leaving the vaults. even today repairmen can convert simple diesel engines (the ones used in irrigation) into basic transportation vehicles.
since people wouldn't have any difficulties in finding engine parts, they would start repairing cars from the day they finished making walls around the cities. because transportation would be urgently needed in search for other people or more fertile areas. and if they had advanced in energy and fusion technology before the great war like in the game, then it would be even easier, for building electric powered motors are much easier than building diesel powered ones.
due to the already existing knowledge on engines and electronics, in 200 years, people would have a more advenced technology than we have today. they would have better cars, better aircrafts, better tanks... they would be building new vaults for The Great War II.