  We present radio observations at 1.5 GHz of 32 local objects selected to
reproduce the physical properties of $z\sim5$ star-forming galaxies. We also
report non-detections of five such sources in the sub-millimetre. We find a
radio-derived star formation rate which is typically half that derived from
H$\alpha$ emission for the same objects. These observations support previous
indications that we are observing galaxies with a young dominant stellar
population, which has not yet established a strong supernova-driven synchrotron
continuum. We stress caution when applying star formation rate calibrations to
stellar populations younger than 100 Myr. We calibrate the conversions for
younger galaxies, which are dominated by a thermal radio emission component. We
improve the size constraints for these sources, compared to previous unresolved
ground-based optical observations. Their physical size limits indicate very
high star formation rate surface densities, several orders of magnitude higher
than the local galaxy population. In typical nearby galaxies, this would imply
the presence of galaxy-wide winds. Given the young stellar populations, it is
unclear whether a mechanism exists in our sources that can deposit sufficient
kinetic energy into the interstellar medium to drive such outflows.
