We present a model for the radio evolution of supernova remnants (SNRs) obtained by using three-dimensional hydrodynamic simulations coupled with nonlinear kinetic theory of cosmic-ray (CR) acceleration in SNRs. We model the radio evolution of SNRs on a global level by performing simulations for a wide range of the relevant physical parameters, such as the ambient density, supernova (SN) explosion energy, acceleration efficiency, and magnetic field amplification (MFA) efficiency. We attribute the observed spread of radio surface brightnesses for corresponding SNR diameters to the spread of these parameters. In addition to our simulations of Type Ia SNRs, we also considered SNR radio evolution in denser, nonuniform circumstellar environments modified by the progenitor star wind. These simulations start with the mass of the ejecta substantially higher than in the case of a Type Ia SN and presumably lower shock speed. The magnetic field is understandably seen as very important for the radio evolution of SNRs. In terms of MFA, we include both resonant and nonresonant modes in our large-scale simulations by implementing models obtained from first-principles, particle-in-cell simulations and nonlinear magnetohydrodynamical simulations. We test the quality and reliability of our models on a sample consisting of Galactic and extragalactic SNRs. Our simulations give Sigma - D slopes between -4 and -6 for the full Sedov regime. Recent empirical slopes obtained for the Galactic samples are around -5, while those for the extragalactic samples are around -4.