So, while it takes place in the future, it is not our future, but a future as imagined in traditional 1950s style science fiction. Note that the Fallout world is not our own, but rather one that has diverged from ours soon after World War II.
However, several songs are described as existing in the Fallout world in the future. It primarily deals with the licensed tracks which were not specifically composed for the Fallout series.
What follows is a timeline of the music associated with the Fallout games based on their recording date.