The cataclysmic explosions of massive stars as supernovae are one of the key ingredients of galaxy formation. However, their evolution is not well understood in the presence of magnetic fields or cosmic rays (CRs). We study the expansion of individual supernova remnants (SNRs) using our suite of 3D hydrodynamical (HD), magnetohydrodynamical (MHD) and CRMHD simulations generated using ramses. We explore multiple ambient densities, magnetic fields, and fractions of supernova energy deposited as CRs (χ_CR), accounting for CR anisotropic diffusion and streaming. All our runs have comparable evolutions until the end of the Sedov-Taylor phase. However, our CRMHD simulations experience an additional CR pressure-driven snowplough phase once the CR energy dominates inside the SNR. We present a model for the final momentum deposited by supernovae that captures this new phase: |$p_{\rm SNR} = 2.87\times 10^{5} (\chi _{\text{CR}} + 1)^{4.82}\left(\frac{n}{\text{cm}^{-3}}\right)^{-0.196} M_{\odot }$| km s^−1. Assuming a 10 per cent fraction of SN energy in CRs leads to a 50 per cent boost of the final momentum, with our model predicting even higher impacts at lower ambient densities. The anisotropic diffusion of CRs assuming an initially uniform magnetic field leads to extended gas and CR outflows escaping from the supernova poles. We also study a tangled initial configuration of the magnetic field, resulting instead in a quasi-isotropic diffusion of CRs and earlier momentum deposition. Finally, synthetic synchrotron observations of our simulations using the polaris code show that the local magnetic field configuration in the interstellar medium modifies the overall radio emission morphology and polarization.