The cosmological redshift drift (Sandage–Loeb effect) offers a unique, model-independent probe of the expansion of the Universe, by directly measuring the temporal fluctuation of cosmic redshift. In this study, we examine the viability of employing the Square Kilometre Array (SKA) to detect the redshift drift signal using neutral hydrogen (HI) 21 cm emission from galaxies and 21 cm absorption from Damped Lyman-α systems. We demonstrate how, by focusing on ultra-high spectral resolutions of 0.001–0.002 Hz and a semi-annual observational baseline, predicted SKA data spanning the redshift range 0 < z < 1 may attain millimeter-per-second sensitivity to the velocity drift. Simulations show that in samples of more than a billion HI-emitting galaxies, velocity drifts can be restricted at the level of 0.05–0.15 cm s⁻¹ every 0.5 yr, whereas HI absorption systems provide complementary but weaker constraints due to their lower number density. Using simulated redshift drift data, we constrain cosmological parameters for a number of dark energy models, resulting in Hubble constant and matter density values that are consistent with existing observations and dark energy equation-of-state parameters close to a cosmological constant. Our findings highlight the ability of SKA HI 21 cm data to directly investigate cosmic acceleration at low redshift, offering a compelling and independent test of dark energy to supplement optical redshift drift investigations at higher redshifts.