We have studied the μJy radio properties of K-selected galaxy populations detected in the Ultra-Deep Survey (UDS) portion of the United Kingdom Infrared Telescope (UKIRT) Deep Sky Survey using 610- and 1400-MHz images from the Very Large Array and the Giant Metre-wave Telescope. These deep radio mosaics, combined with the largest and deepest K-band image currently available, allow high signal-to-noise ratio (S/N) detections of many K-selected subpopulations, including sBzK and pBzK star-forming and passive galaxies. We find a strong correlation between the radio and K-band fluxes and a linear relationship between star formation rate (SFR) and K-band luminosity. We find no evidence, either from radio spectral indices or a comparison with submm-derived SFRs, that the full sample is strongly contaminated by active galactic nuclei (AGN) at these low flux densities, though this is very difficult to determine from this data set. The photometric redshift distributions for the BzK galaxies place 37 (29) per cent of pBzK (sBzK) galaxies at z <1.4, implying that location on the BzK diagram alone is not sufficient to select samples at 1.4 <z <2.5. The sBzK and pBzK galaxies have similar levels of radio flux density, SFR and specific SFR (SSFR - SFR per unit stellar mass) at z <1.4, suggesting there is strong contamination of the pBzK sample by star-forming galaxies. At z > 1.4, the pBzK galaxies become difficult to detect in the radio stack, though the implied SFRs are still much higher than expected for passively evolving galaxies. It may be that their radio emission comes from low-luminosity AGN. Extremely red objects (EROs) straddle the passive and star-forming regions of the BzK diagram and also straddle the two groups in terms of their radio properties. We find that K-bright ERO samples are dominated by passive galaxies and faint ERO samples contain more star-forming galaxies. The star formation history (SFH) from stacking all K-band sources in the UDS agrees well with that derived for other wavebands and other radio surveys, at least out to z ~ 2. The radio-derived SFH then appears to fall more steeply than that measured at other wavelengths. The SSFR for K-selected sources rises strongly with redshift at all stellar masses, and shows a weak dependence on stellar mass. High- and low-mass galaxies show a similar decline in SSFR since z ~ 2.