So, there have been more than a couple reasons for Fallout to be on my mind, and that includes the unique soundtrack they have in game. Although this was fleshed out more in Fallout 3 and Fallout: New Vegas, classic music from the 1940s through 1960s is a dominant theme in Fallout.
This makes me wonder where music may have went for the 1970s through the mid 21st century. Could "newer" genres like heavy metal be invented, even with the vastly different history of the Fallout world? On the other hand, is it possible that the in game music we are hearing, even though it is from the 1940s through 1960s in our world, is from the 2040s through 2070s in the Fallout world?
We do have an example of post-nuclear music being made, at least, in New Vegas with the Lonesome Drifter and other performers at The Tops. This indicates to me that acoustic oriented non electric music would be the most common new stuff to come out - there isn't exactly much room for a bustling scene. I would be interested to see if such performers are seen in Fallout 4, given Bethesda Game Studios incorporation of tavern minstrels in Skyrim.
So, I think there was a Fallout world equivalent to Black Sabbath, that may have even led to heavier forms of metal like Judas Priest, but it might not have gotten quite as far as our world did, maybe no heavier than the first wave of thrash metal bands, if even that. The technology to produce such sounds did not depend on new technology, though. Being that there is still a century between the time Black Sabbath started (if they existed at all) and the final decade of normalcy, it is also likely such music may have been forgotten. Or maybe not, maybe it really is underground and not really played on the post-nuclear radio stations.
There are also signs some sort of punk rock may have existed - many raiders sport hair styles that would fit in with the punk scene (mohawks and the like). Perhaps such music is seen to be unfit for society and is solely enjoyed by the outcasts.
Microchip dependent music would never have become dominant. There are indications that Fallout's world may have had its own 1980s type era in the 2030s, with video games becoming common at that time as well.