We report on deep near-infrared observations obtained with the Wide Field Camera-3 (WFC3) onboard the Hubble Space Telescope (HST) of the first five confirmed gravitational lensing events discovered by the Herschel Astrophysical Terahertz Large Area Survey (H-ATLAS). We succeed in disentangling the background galaxy from the lens to gain separate photometry of the two components. The HST data allow us to significantly improve on previous constraints of the mass in stars of the lensed galaxy and to perform accurate lens modelling of these systems, as described in the accompanying paper by Dye et al. We fit the spectral energy distributions of the background sources from near-IR to millimetre wavelengths and use the magnification factors estimated by Dye et al. to derive the intrinsic properties of the lensed galaxies. We find these galaxies to have star-formations rates (SFR) ~ 400-2000 M⊙ yr-1, with ~(6-25) × 1010 M⊙ of their baryonic mass already turned into stars. At these rates of star formation, all remaining molecular gas will be exhausted in less than ~100 Myr, reaching a final mass in stars of a few 1011 M⊙. These galaxies are thus proto-ellipticals caught during their major episode of star formation, and observed at the peak epoch (z ~ 1.5-3) of the cosmic star formation history of the Universe.