Dispersion laws of the full optical phonon spectrum of a nanotube made of würtzite-type materials as well as the corresponding Fröhlich electron-phonon interaction terms are derived and studied within the framework of the dielectric continuum model for a uniaxial crystal. The coupling coefficients describing electron-phonon interaction are obtained in an analytical closed form, depending on the dispersion law of the involved phonon branch. We present and discuss results of numerical calculations of optical phonon spectrum and Fröhlich coupling coefficients for some chosen würtzite AlN nanotubes. Observed features, induced by the anisotropy of würtzite-type materials, are discussed.