General expressions of the electron-phonon Hamiltonians in ring-like nanostructures are settled. A unified macroscopic continuum approach for the treatment of acoustic and optical phonon modes in doubly-connected nanowires is established. A basis for the space of solutions is derived, and by applying the appropriate boundary conditions, the dispersion relation curves, as well as the displacement fields and the electrostatic potential for non-polar and polar optical modes are reported. Employing the methods of quantum field theory, the electron- and hole-zone-center optical phonon deformation potential and optical long-range as well as electron-acoustical phonon Hamiltonians are deduced in a systematic way. The results are valid for the study of polar and non-polar semiconductor based core-shell nanowires and to analyze the role of intrinsic strain and the geometric factors on the electron-phonon coupling strengths. Special emphasis is placed on the importance of the cylindrical symmetry in the interaction Hamiltonians.