The Great Mushroom War is the cause of the post-apocalyptic condition of the Land of Ooo. It is thought to have been fought with nuclear war-heads and wiped out all of humanity (except for Finn). It also blew a chunk of the Earth off. The title probably refers to the mushroom shaped clouds made by the bombing. In "Finn the Human," TGMW played a large role. It is proven that The Lich created it, and that without it, nothing would be the same. On a similar note, in "Simon and Marcy," which takes place a couple years after the war, the city is inhabited by somewhat humanoid slime monster which could suggest that the bomb mutated the inhabitants of the city. This also can explain why there was that was of bubblegum that was alive and gave Simon some chicken soup. Obviously, it was once a human. This might also explain why Ooo has all of those weird creatures.
Ice King - Simon Petrikov was saved from Nuclear Winter by his crown, because he could survive cold temperatures.
Marceline - Being a vampire with heightened abilities, Marceline managed to survive.
Hunson Abadeer - Marceline's dad was a demon, so he could survive.
Finn (Possibly) - Finn or Finn's ancestors could have possibly survived the war.
Lady Rainicorn (Possibly) - Lady's age is unknown, so she could be old enough to have been alive when the war happened, and survived.
Princess Bubblegum (Possibly) - A pink goo seen in "Simon and Marcy" could be her due to signs and clues throughout the series. Although it could be her ancestors.