A family of models of counterrotating and rotating relativistic thin discs of infinite extension based on a charged and magnetized Kerr-NUT metric are constructed using the well-known "displace, cut and reflect" method extended to solutions of vacuum Einstein-Maxwell equations. The metric considered has as limiting cases a charged and magnetized Taub-NUT solution and the well known Kerr-Newman solutions. We show that for Kerr-Newman fields the eigenvalues of the energy-momentum tensor of the disc are for all the values of the parameters real quantities so that these discs do not present heat flow in any case, whereas for charged and magnetized Kerr-NUT and Taub-NUT fields we find always regions with heat flow. We also find a general constraint over the counterrotating tangential velocities needed to cast the surface energy-momentum tensor of the disc as the superposition of two counterrotating charged dust fluids. We show that, in general, it is not possible to take the two counterrotating fluids as circulating along electrogeodesics nor take the two counterrotating tangential velocities as equal and opposite.