This is actually a more substantial POD than most people might assume, especially for the 80s and 90s immediately. There were several genres in the 80s and 90s that owed quite a bit of their existence to Hip Hop/ Rap. I'll only touch on the urban side of things for now.
No Hip Hop likely means no Electro (aka Breakdancing Music), which in turn means no Latin Freestyle or Techno, which both evolved from Electro. Basically, anything with a "breakbeat" goes away. There's also no New Jack Swing in the late 80s since that evolved from a later version of Hip Hop. Needless to say, the entire Hip Hop culture and music of the 90s to the present doesn't happen either.
So what takes its place? Well for starters, House Music, which evolved from Disco should do well, perhaps remaining mainstream till the present day. The biggest beneficiary though will probably be Prince, of all people. Prince's music was always almost devoid of contemporary influences, like Hip Hop. Prince was essentially making 70s Funk with an 80s veneer, uninformed by the prevailing 80s musical trends (except the synthesizer). In the complete absence of New York Hip Hop and Electro inflected R&B, Prince's Minneapolis sound should dominate the 80s. Stuff like Go Go music out of the DC area might take the "cool urban underground" mantle that Hip Hop occupied in OTL. Lastly, an obscure genre like Acid Jazz might gain more mainstream acceptance in the 90s as a more modern sophisticated take on the Minneapolis sound of the 80s.
What would that all mean for the present day, who knows?