The timescape cosmology represents a potentially viable alternative to the standard homogeneous cosmology, without the need for dark energy. Although average cosmic evolution in the timescape scenario only differs substantially from that of Friedmann-Lemaˆıtre model at relatively late epochs when the contribution from the energy density of radiation is negligible, a full solution of the Buchert equations to incorporate radiation is necessary to smoothly match parameters to the epoch of photon decoupling and to obtain constraints from cosmic microwave background data. Here we extend the matter-dominated solution found in earlier work to include radiation, providing series solutions at early times and an efficient numerical integration strategy for generating the complete solution. The numerical solution is used to directly calculate the scale of the sound horizon at decoupling, and at the baryon drag epoch. The constraints on these scales from the Planck satellite data yield bounds on the timescape cosmological parameters, which are found to also agree with the best-fit values from a recent analysis of SDSS-II supernova data, while avoiding the problem of a primordial lithium-7 abundance anomaly.