If the nuclear holocaust happened in 2077, then why are the clothes and music in the style of the 1950s? (source)
Because the Fallout universe is set in an alternate history. Basically the Fallout universe explores how the world could have evolved if the nuclear visions of the 50's would have come true. It's called Retro-futurism, and it's been executed perfectly by the game designers IMHO.
As you can tell from the background info you get in the game world, the society in the year 2077 has stopped evolving, its ideals of family, patriotism, life style etc. come straight out of the 50's. The design of cars, planes, buildings, and the technology behind are the exact same as envisioned in our 50's, except in the Fallout universe, they've actually come true. If you've played Fallout 3,
even the aliens and their death ray weapons are exactly like imagined back then.