GRB 130427A was extremely bright as a result of occurring at low redshift whilst the energetics were more typical of high-redshift gamma-ray bursts (GRBs). We collected well-sampled light curves at 1.4 and 4.8 GHz of GRB 130427A with the Westerbork Synthesis Radio Telescope (WSRT); and we obtained its most accurate position with the European Very Long Baseline Interferometry Network (EVN). Our flux density measurements are combined with all the data available at radio, optical and X-ray frequencies to perform broad-band modelling in the framework of a reverse–forward shock model and a two-component jet model, and we discuss the implications and limitations of both models. The low density inferred from the modelling implies that the GRB 130427A progenitor is either a very low metallicity Wolf–Rayet star, or a rapidly rotating, low-metallicity O star. We also find that the fraction of the energy in electrons is evolving over time, and that the fraction of electrons participating in a relativistic power-law energy distribution is less than 15 per cent. We observed intraday variability during the earliest WSRT observations, and the source sizes inferred from our modelling are consistent with this variability being due to interstellar scintillation effects. Finally, we present and discuss our limits on the linear and circular polarization, which are among the deepest limits of GRB radio polarization to date.