  Aims. We investigate the role played by a pre-supernova (SN) ambient magnetic
field on 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
model 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
by 3D MHD 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. From the simulations, we
synthesize both thermal X-ray and non-thermal radio emission and compare the
model results with observations. Results. The presence of an ambient 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 years). 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. 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.
