We have seen Star Fleet use NX for experimental vessels before.
We haven't seen Star Fleet use class names as part of the ship's registry.
That's because, as has already been stated, there's the Earth Starfleet, and then there's the Federation Starfleet.
In the Earth Starfleet, the Enterprise, Columbia, and presumably the four other ships named after the space shuttles are of the NX class. Presumably past Earth Starfleet vessels were of such classes as the NU, NV, and NW classes, and that ships after the NX class might have been designated as NY or NZ class. The class name was followed by a production number, which made up the ship's registry. But that's just speculation on my part, because the show never provided any cut-and-dried information about that, other than the NX-01 Enterprise and the NX-02 Columbia.