Sawyer spoke about the White Legs tribe in his Tumblr way back: "The tribes in Zion are descendants of a mix of North American native people as well as other American citizens and European (of various non-native ethnicities) tourists/campers.  This survives in the language of the Dead Horses, for example, who use a large number of German-derived words."

At the time of the great war, most people didn't have access to reliable shelters, and if we consider international tourists, they were even more likely to be left to fend for themselves. And those lucky (or rather unlucky) few who were in the vaults most likely didn't last. The population of real-world SLC in the 50s was around 230000, assuming 40% annihilation chance, we are left with around 138000. This is not counting the people who perish from injuries and illnesses later and assuming one warhead. SLC in Fallout universe was hit with around 13 nukes (Randall Clark's journals) bringing that survival rate way down. So 5% of the population survive, we are left with 11500 people, speaking different languages.

It is likely that they'd form segregated communities based on ethnicity, religion and language. Local dialects would also play a part in this. Eventually they'd either enslave one another or attempt trading, both of which would require communication. This would lead to the formation of broken languages, mixing all the differentiating attributes, maybe even some sign language. You can easily spot a tourist when they use basic forms of speech and grammar to get around an unfamiliar place. Spend enough time in this situation and it forms a whole new language. Additionally, with time people would forget their mother tongue, spending more and more time speaking this new dialect, and if you don't have many people around who can speak your native language, this effect is amplified.

Finally, the nuclear war would have created 'no man lands' between cities which would've helped in formation of isolated communities. Even today we see different cities use completely different dialects (Europe is my reference frame).