A simple rigorous analytical theory of two-dimensional (2D) nonequilibrium electrons occupying an arbitrary number of subbands in a quantum well is developed. The electric-field dependence of electron mobility and the average kinetic energy for AlN/GaN quantum wells are presented. At temperatures below 200 K the electron mobility is controlled mainly by the acoustic phonon scattering and it is a nonmonotonous function of the electric field, which has a maximum. At room and higher temperatures the interaction with both acoustic and polar optical phonons determine the hot-electron mobility and it depends very weakly on the electric field. Both the mobility and average energy of 2D electrons are smaller than that for three-dimensional (3D) electrons in the bulk semiconductor. Our theory provides a self-consistent transition from the 2D to the 3D regime of electron transport with increasing electric field accompanied by the occupation of an increasingly large number of subbands by the electrons.