Rap-metal or Rap-Rock. simple.

Yeah, Rap-metal or rap-rock. Though your suggestion of nu metal works as well.

LP sort of became Nu metal after the fact. When they came, the genre either wasn't widespread or well-known enough for them to be put into it, but they do fit into the category, for the most part anyway.

There are more factors that go into it than simply a mash up of rock and rap. What kind of rock? what kind of rap? Gym Class Heroes, I kind of want to say .... alt/pop with rap or R&B roots or something like that, whereas LP (simply for sake of example) also combine rap and rock elements, but create a completely different sound.

Long story short, there's more to it than simply rock and rap, and it's possibly that there are no genre tags available to fully explain what they are.