The authors theoretically investigated the electron mobility in the nanometer thickness AlNGaNAlN heterostructures limited by the polar optical and confined acoustic phonons. The proposed model accurately takes into account dispersion of the optical and acoustic phonons in such heterostructures as well as inelasticity of the electron scattering on both optical and acoustic phonons. It has been shown that the intersubband electronic transitions play an important role in limiting the electron mobility when the energy separation between one of the size-quantized excited electron subbands and the Fermi energy becomes comparable to the optical or confined acoustic phonon energy. The latter results in the nonmonotonic oscillatory dependence of the electron mobility on the thickness of the GaN conduction channel layer. The predicted effect is observable at room temperature and over a wide range of carrier densities. The described mechanism can be used for fine tuning the confined electron and phonon states in the nanoscale heterostructures made of different material systems in order to achieve performance enhancement of the nanoscale electronic devices.