The film you are looking for may be
A Boy and His Dog.
Also, there are already multiple threads on this subject, so please use the search function in the future.
And another thing. The word apocalypse has nothing to do with a nuclear catastrophe, and I keep seeing folks assume that it does. For the record:
Main Entry: apoc·a·lypse
Pronunciation: &-'pä-k&-"lips
Function: noun
Etymology: Middle English, revelation, Revelation, from Late Latin apocalypsis, from Greek apokalypsis, from apokalyptein to uncover, from apo- + kalyptein to cover -- more at HELL
1 a : one of the Jewish and Christian writings of 200 B.C. to A.D. 150 marked by pseudonymity, symbolic imagery, and the expectation of an imminent cosmic cataclysm in which God destroys the ruling powers of evil and raises the righteous to life in a messianic kingdom b capitalized : REVELATION 3
2 a : something viewed as a prophetic revelation b : ARMAGEDDON
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Main Entry: Ar·ma·ged·don
Pronunciation: "är-m&-'ge-d&n
Function: noun
Etymology: Greek ArmageddOn, HarmagedOn, scene of the battle foretold in Revelations 16:14-16
1 a : the site or time of a final and conclusive battle between the forces of good and evil b : the battle taking place at Armageddon
2 : a usually vast decisive conflict or confrontation
As you can see, nuclear disaster is not mentioned anywhere in these definitions, although Armegeddon more closely dictates a decisive conflict, although nuclear weapons do not have to be involved at all to fit this definition. So, if the films you are looking for are
post-nuclear, then I would check out the films listed by these other kind folks, as well as
Hiroshima.