Yup according to the
Mad Max 2
story-notes written by George Miller and Terry Hayes (which can be found on the Internet), the nuclear war and final civilization-collapse don't occur until after Max has been in the wasteland for several months; up until that point, Australian society is still humming along, if starting to finally come apart bit by bit (as we see demonstrated in the first film).

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I just watch the film for the first time too. And i would say that its like thing were F##ked up. But not f##ked up enough that you couldn't take your family on a road trip. It felt like everything was really sh!tty than full on dystopian. I haven't seen Road Warrior in a long time. But i don't remember if it explains how thing got even worse.