Resolved maps of the thermal and non-thermal radio continuum (RC) emission of distant galaxies are a powerful tool for understanding the role of the interstellar medium (ISM) in the evolution of galaxies. We simulate the RC surface brightness of present-day star-forming galaxies in the past at 0.15 < z < 3, considering two cases of radio size evolution: (1) no evolution and (2) same evolution as in the optical. We aim to investigate the (a) structure of the thermal and non-thermal emission on kpc scales, (b) evolution of the thermal fraction and synchrotron spectrum at mid-radio frequencies (≃1-10 GHz), and (c) capability of the proposed SKA phase 1 MID-frequency (SKA1-MID) reference surveys in detecting the RC emitting structures. The synchrotron spectrum flattens with z causing curvature in the observed mid-radio SEDs of galaxies at higher z. The spectral index reported in recent observational studies agrees better with the no size evolution scenario. In this case, the mean thermal fraction observed at 1.4 GHz increases with redshift by more than 30 per cent from z = 0.15 to z = 2 because of the drop of the synchrotron emission at higher rest-frame frequencies. More massive galaxies have lower thermal fractions and experience a faster flattening of the non-thermal spectrum. The proposed SKA1-MID band 2 reference survey, unveils the ISM in M51- and NGC 6946-like galaxies (with ${\rm M_{\star }}\simeq 10^{10}\, {\rm M}_{\odot }$) up to z = 3. This survey detects lower-mass galaxies like M33 (${\rm M_{\star }}\simeq 10^{9}\, {\rm M}_{\odot }$) only at low redshifts z ≲ 0.5. For a proper separation of the RC emitting processes at the peak of star formation, it is vital to include band 1 into the SKA1-MID reference surveys.