Abstract Interferometry of the cosmic 21-cm signal is set to revolutionise our understanding of the Epoch of Reionisation (EoR) and the Cosmic Dawn (CD). The culmination of ongoing efforts will be the upcoming Square Kilometre Array (SKA), which will provide tomography of the 21-cm signal from the first billion years of our Universe. Using a galaxy formation model informed by high-z luminosity functions, here we forecast the accuracy with which the first phase of SKA-low (SKA1-low) can constrain the properties of the unseen galaxies driving the astrophysics of the EoR and CD. We consider three observing strategies: (i) deep (1000h on a single field); (ii) medium-deep (100hr on 10 independent fields); and (iii) shallow (10hr on 100 independent fields). Using the 21-cm power spectrum as a summary statistic, and conservatively only using the 21-cm signal above the foreground wedge, we predict that all three observing strategies should recover astrophysical parameters to a fractional precision of ∼0.1 – 10 per cent. The reionisation history is recovered to an uncertainty of $Δ z \mathrel {\lesssim}0.1$ (1σ) for the bulk of its duration. The medium-deep strategy, balancing thermal noise against cosmic variance, results in the tightest constraints, slightly outperforming the deep strategy. The shallow observational strategy performs the worst, with up to a ∼10 – 60 per cent increase in the recovered uncertainty. We note, however, that non-Gaussian summary statistics, tomography, as well as unbiased foreground removal would likely favour the deep strategy.