EXPLANATION #1
The Great Mushroom War wasn't fought with nuclear bombs
It was fought with the "C-bombs" from Calvin and Hobbes. To clarify, C-bombs were a throwaway gag in one strip, where they were said to be destructive Fat Man-size projectiles of Calvin's imagination. It stands to reason that a global war fought with these things would result in almost everything gaining life, sentience, and intelligence. Finn and Jake are the reincarnations of Calvin and Hobbes, and share their dynamic. Because of the fact that no other residents of Calvin's hometown have gone on, or at least haven't appeared yet, Finn and Jake don't have any memories of a past life other than being friends and exploring the world around them. However, several people were [[Flanderization Flanderized]] from their pre-war personalities into magical and powerful beings: Marceline was probably an Emo Teen who was cosmically retconned into being an actual vampire, her dad was a Corrupt Corporate Executive who ordered the C-bombings in the first place, Lumpy Space Princess was a spoiled teen experimenting with drugs, and so on.
EXPLANATION #2
The Great Mushroom War was fought over literal mushrooms.
When the magic came back, probably due to the reawakening of the Cosmic Owl, it came in the form of new species of mushroom growing all over the world. When they grew in great numbers the land around them became a wellspring of magic, creating the various microcosoms of Ooo- for example, a species of Ice Mushroom grew where he Ice Kingdom now sits, and that's what allows even normal folks like Finn to perform ice-ninja magic there. Localized anomalies, such as talking mountains and superpower-granting mud puddles, also came into being. Those who ate these mushrooms took on their attributes as well, resulting in populations of candy-people, blue warrior people, nut-people, and so on. The war over these mushrooms caused much of normal civilization to collapse, but for the most part society simply mutated into the kingdoms we see in Ooo today. Finn is capable of such heroism despite his lack of magical powers specifically because he's among the last remaining humans. This is also why so many of the show's females are so into him despite his youth, awkwardness, and lack of gainful employment- he can theoretically produce viable offspring with any of them.
Source
http://tvtropes.org/pmwiki/pmwiki.php/WMG/AdventureTimeBoth are pretty interesting ideas, but im leaning towards the first one with the C-bomb