The self-charging of radioactive aerosols will be reduced by background ions, such as those produced by radioactive gases. The sources of these background ions and their production rates are specified for a reactor containment atmosphere during a possible nuclear accident. Previous theory is extended to calculate the charging of a polydisperse radioactive aerosol. Gaussian approximations to charge distributions on an aerosol of a given size, and are shown to give a good representation of the exact numerical charge distributions of a Cs aerosol at normal temperatures, and also for highly radioactive aerosol containing I-131 in a containment atmosphere. Extensive calculations are performed for charge-induced modifications to Brownian coagulation rates between steady-state size distributions of these radioactive aerosols, and also between small-sized radioactive aerosol and larger (non-radioactive) aerosol. The results show considerable enhancements of the coagulation rates between large and small-sized aerosol, but also a strong suppression of coagulation between large particles. Rate modifications calculated using the Gaussian approximations are generally close to the exact values. Time-dependent calculations for a monodisperse cc-decaying aerosol reveal enhancements in coagulation rates even when the average charge on the aerosol is positive. Our results are relevant to behaviour in a dusty plasma.