It is important to remember that Languages would change even without the fall of civilisation as we know it. Thus Languages would change at an even faster pace than normally due to the collapse of Governments and an increase of violent societies.
The closest thing to Nuclear war that we know about is the Black Death where the collapse of society created New Languages.
I'm currently watching a TV series, The 100, in which the descendants of the survivors of a nuclear war, who've been living in a space station for nearly a century, come back to Earth and find that the inhabitants of what is now the Washington D.C. area have reverted to a hunter-gatherer level of development, and have developed a language, descended from American English, that is not mutually intelligible with it.
The language's author (the same guy who made Dothraki) said that the language originated as a jargon spoken by the warriors in order to not be understood by their enemies but, I find it very hard to believe that an entire people can be nuked back to the Stone Age and develop a brand new language in barely more than three generations.
It's a good show nonetheless, though; the first season is just decent (it felt too much like a Battlestar Galactica/Lord of the Flies hybrid) but the second and third seasons are the closest thing anyone's ever done to a Fallout TV series. The Battlestar Galactica vibes are coming back though, since Jaha's become the second coming of Gaius Baltar.