We complement our previous analysis of a sample of z similar to 1-2 luminous and ultraluminous infrared galaxies [(U)LIRGs], by adding deep Very Large Array radio observations at 1.4 GHz to a large data set from the far-UV to the submillimetre, including Spitzer and Herschel data. Given the relatively small number of (U)LIRGs in our sample with high signal-to-noise (SIN) radio data, and to extend our study to a different family of galaxies, we also include six well-sampled near-infrared (near-IR)-selected BzK galaxies at z similar to 1.5. From our analysis based on the radtran spectral synthesis code GRASIL, we find that, while the IR luminosity may be a biased tracer of the star formation rate (SFR) depending on the age of stars dominating the dust heating, the inclusion of the radio flux offers significantly tighter constraints on SFR. Our predicted SFRs are in good agreement with the estimates based on rest-frame radio luminosity and the Bell calibration. The extensive spectrophotometric coverage of our sample allows us to set important constraints on the star formation (SF) history of individual objects. For essentially all galaxies, we find evidence for a rather continuous SFR and a peak epoch of SF preceding that of the observation by a few Gyr. This seems to correspond to a formation redshift of z similar to 5-6. We finally show that our physical analysis may affect the interpretation of the SFR-M-* diagram, by possibly shifting, with respect to previous works, the position of the most dust obscured objects to higher M-*, and lower SFRs.