Our universe and the Fallout universe didn't diverge from eachother until the late 1950's. That's why the opening of the game talks about real-world events, like the bombings of Hiroshima and Nagasaki. At some point, differences start to appear, but even then there's never a clean line where the two diverge. In our world, Yuri Gagarin was the first human being in outer space. In the Fallout world, it was Carl Bell. Other things stay the same, like the Hippie movement appearing during the '60's.
So the artists who recorded those songs were alive during the 1940's, 50's and 60's, both in our world and in the Fallout world. Their 2077 isn't our 1950's, because in the 1950's we didn't have Power Armor, sentient robots or nuclear-fueled cars either. Their 2077 is our 2077, just radically different.