Aims. We investigate the role played by a pre-supernova (SN) ambient magnetic field in the dynamics of the expanding remnant of SN 1987A, and the origin and evolution of the radio emission from the remnant, in particular during the interaction of the blast wave with the nebula surrounding the SN. Methods. We modeled the evolution of SN 1987A from the breakout of the shock wave at the stellar surface to the expansion of its remnant through the surrounding nebula using three-dimensional magnetohydrodynamic simulations. The model considers the radiative cooling, the deviations from equilibrium of ionization, the deviation from temperature-equilibration between electrons and ions, and a plausible configuration of the pre-SN ambient magnetic field. We explore the strengths of the pre-SN magnetic field ranging between 1 and 100 μG at the inner edge of the nebula and we assume an average field strength at the stellar surface B₀ ≈ 3 kG. From the simulations, we synthesize the thermal X-ray and the non-thermal radio emission and compare the model results with observations. Results. The presence of an ambient magnetic field with strength in the range considered does not change significantly the overall evolution of the remnant. Nevertheless, the magnetic field reduces the erosion and fragmentation of the dense equatorial ring after the impact of the SN blast wave. As a result, the ring survives the passage of the blast, at least during the time covered by the simulations (40 yr). Our model is able to reproduce the morphology and lightcurves of SN 1987A in both X-ray and radio bands. The model reproduces the observed radio emission if the flux originating from the reverse shock is heavily suppressed. In this case, the radio emission originates mostly from the forward shock traveling through the H II region and this may explain why the radio emission seems to be insensitive to the interaction of the blast with the ring. Possible mechanisms for the suppression of emission from the reverse shock are investigated. We find that synchrotron self-absorption and free–free absorption have negligible effects on the emission during the interaction with the nebula. We suggest that the emission from the reverse shock at radio frequencies might be limited by highly magnetized ejecta.