Cosmological reheating describes the transition of the postinflationary universe to a hot and thermal state. In order to shed light on the underlying dynamics of this process, we propose to quantum-simulate the reheating-like dynamics of a generic cosmological single-field model in an ultracold Bose gas. In our setup, the excitations on top of an atomic Bose-Einstein condensate play the role of the particles produced by the decaying inflaton field after inflation. Expanding spacetime as well as the background oscillating inflaton field are mimicked in the nonrelativistic limit by a time dependence of the atomic interactions, which can be tuned experimentally via Feshbach resonances. As we illustrate by means of classical-statistical simulations for the case of two spatial dimensions, the dynamics of the atomic system exhibits the characteristic stages of far-from-equilibrium reheating, including the amplification of fluctuations via parametric instabilities and the subsequent turbulent transport of energy towards higher momenta. The transport is governed by a nonthermal fixed point showing universal self-similar time evolution as well as a transient regime of prescaling with time-dependent scaling exponents. While the classical-statistical simulations can capture only the earlier stages of the dynamics for weak couplings, the proposed experiment has the potential of exploring the evolution up to late times even beyond the weak coupling regime.