A complete, flux density limited sample of 96 faint ($> 0.5$ mJy) radio sources is selected from the 10C survey at 15.7 GHz in the Lockman Hole. We have matched this sample to a range of multi-wavelength catalogues, including SERVS, SWIRE, UKIDSS and optical data; multi-wavelength counterparts are found for 80 of the 96 sources and spectroscopic redshifts are available for 24 sources. Photometric reshifts are estimated for the sources with multi-wavelength data available; the median redshift of the sample is 0.91 with an interquartile range of 0.84. Radio-to-optical ratios show that at least 94 per cent of the sample are radio loud, indicating that the 10C sample is dominated by radio galaxies. This is in contrast to samples selected at lower frequencies, where radio-quiet AGN and starforming galaxies are present in significant numbers at these flux density levels. All six radio-quiet sources have rising radio spectra, suggesting that they are dominated by AGN emission. These results confirm the conclusions of Paper I that the faint, flat-spectrum sources which are found to dominate the 10C sample below $∼ 1$ mJy are the cores of radio galaxies. The properties of the 10C sample are compared to the SKADS Simulated Skies; a population of low-redshift starforming galaxies predicted by the simulation is not found in the observed sample. ; Comment: 21 pages, 24 figures, accepted by MNRAS