We have used a spin-wave model to calculate the temperature dependence of the (sublattice) magnetization of magnetic nanoparticles. The uniform precession mode, corresponding to a spin wave with wave vector q=0, is predominant in nanoparticles and gives rise to an approximately linear temperature dependence of the (sublattice) magnetization well below the superparamagnetic blocking temperature for both ferro-, ferri-, and antiferromagnetic particles. This is in accordance with the results of a classical model for collective magnetic excitations in nanoparticles. In nanoparticles of antiferromagnetic materials, quantum effects give rise to a small deviation from the linear temperature dependence of the (sublattice) magnetization at very low temperatures. The complex nature of the excited precession states of nanoparticles of antiferromagnetic materials, with deviations from antiparallel orientation of the sublattice magnetization vectors, results in a contribution to the susceptibility, which increases with increasing temperature.