Energy spectra and the magnetic moments of semiconductor core-shell nanocylinders, subjected to a magnetic field, are theoretically studied for different numbers of electrons in the shell and different height-to-radius ratios of the cylinder. The electron-electron interaction is taken into account within the Hartree-Fock approximation. We focus on the perspectives to experimentally detect the Aharonov-Bohm oscillations in the magnetic moment of multielectron core-shell nanowires. Among the different factors that influence the shape and magnitude of these oscillations, the destructive effect due to the interaction of electrons with randomly distributed charged impurities appears most important. Nevertheless, Aharonov-Bohm oscillations of a sufficiently high magnitude survive when averaged over an assembly of nanowires with different impurity distributions.