As far as I understand, nihilism and existentialism were both reactions to the "death of religion". Nihilism concludes that everything is devoid of meaning. Existentialism concludes basically the same thing, but then is like "so let's go be ourselves and have fun, find ways to enjoy life" basically. I think this is where the "is permitted" part comes in.

But anyway nihilism is related to religion in the sense that it's a reaction to the end of religion.