We have worked out predictions for the radio counts of star-forming galaxies down to nJy levels, along with redshift distributions down to the detection limits of the phase 1 Square Kilometer Array MID telescope (SKA1-MID) and of its precursors. Such predictions were obtained by coupling epoch dependent star formation rate (SFR) functions with relations between SFR and radio (synchrotron and free-free) emission. The SFR functions were derived taking into account both the dust obscured and the unobscured star-formation, by combining far-infrared (FIR), ultra-violet (UV) and H_alpha luminosity functions up to high redshifts. We have also revisited the South Pole Telescope (SPT) counts of dusty galaxies at 95\,GHz performing a detailed analysis of the Spectral Energy Distributions (SEDs). Our results show that the deepest SKA1-MID surveys will detect high-z galaxies with SFRs two orders of magnitude lower compared to Herschel surveys. The highest redshift tails of the distributions at the detection limits of planned SKA1-MID surveys comprise a substantial fraction of strongly lensed galaxies. We predict that a survey down to 0.25 microJy at 1.4 GHz will detect about 1200 strongly lensed galaxies per square degree, at redshifts of up to 10. For about 30% of them the SKA1-MID will detect at least 2 images. The SKA1-MID will thus provide a comprehensive view of the star formation history throughout the re-ionization epoch, unaffected by dust extinction. We have also provided specific predictions for the EMU/ASKAP and MIGHTEE/MeerKAT surveys.