Many technical and biomedical applications of magnetic nanoparticles rely on their response to time-varying magnetic fields. While well-established models exist for either immobile or thermally blocked nanoparticles, the intermediate regime where Brownian as well as Néel relaxation occur at the same time is less well explored. Here, we use an efficient model that allows us to study the nonlinear dynamics of individual magnetic nanoparticles in response to different time-varying magnetic fields over a broad range of field parameters, taking into account both relaxation mechanisms. We provide quasiexact solutions for the longitudinal dynamics as well as approximate formulas from dynamic mean-field theory. Our results are relevant, e.g., for magnetorelaxometry, magnetic fluid hyperthermia, and magnetic particle imaging. For these example applications, we show that the ratio of characteristic Brownian to Néel relaxation time can have a profound impact on characteristic response quantities, especially at large field strengths.