The recent transient event Swift J1644+57 has been interpreted as emission from a collimated relativistic jet, powered by the sudden onset of accretion on to a supermassive black hole following the tidal disruption of a star. Here we model the radio–microwave emission as synchrotron radiation produced by the shock interaction between the jet and the gaseous circumnuclear medium (CNM). At early times after the onset of the jet (t≲ 5–10 d) a reverse shock propagates through and decelerates the ejecta, while at later times the outflow approaches the Blandford–McKee self-similar evolution (possibly modified by additional late energy injection). The achromatic break in the radio light curve of Swift J1644+57 is naturally explained as the transition between these phases. We show that the temporal indices of the pre- and post-break light curve are consistent with those predicted if the CNM has a wind-type radial density profile n∝r−2. The observed synchrotron frequencies and self-absorbed flux constrain the fraction of the post-shock thermal energy in relativistic electrons εe≈ 0.03–0.1, the CNM density at 1018 cm n18≈ 1–10 cm−3 and the initial Lorentz factor Γj≈ 10–20 and opening angle of the jet. Radio modelling thus provides robust independent evidence for a narrowly collimated outflow. Extending our model to the future evolution of Swift J1644+57, we predict that the radio flux at low frequencies (ν≲ few GHz) will begin to brighten more rapidly once the characteristic frequency νm crosses below the radio band after it decreases below the self-absorption frequency on a time-scale of months (indeed, such a transition may already have begun). Our results demonstrate that relativistic outflows from tidal disruption events provide a unique probe of the conditions in distant, previously inactive galactic nuclei, complementing studies of normal active galactic nuclei.