We present a comprehensive set of forecasts for the cross-correlation signal between 21 cm intensity mapping and galaxy redshift surveys. We focus on the data sets that will be provided by the SKAO for the 21 cm signal, DESI and Euclid for galaxy clustering. We build a likelihood which takes into account the effect of the beam for the radio observations, the Alcock-Paczynski effect, a simple parametrization of astrophysical nuisances, and fully exploit the tomographic power of such observations in the range z = 0.7-1.8 at linear and mildly non-linear scales (k < 0.25h Mpc-1). The forecasted constraints, obtained with Monte Carlo Markov Chains techniques in a Bayesian framework, in terms of the six base parameters of the standard ΛCDM model, are promising. The predicted signal-to-noise ratio for the cross-correlation can reach ∼50 for z ∼1 and k ∼0.1h Mpc-1. When the cross-correlation signal is combined with current Cosmic Microwave Background (CMB) data from Planck, the error bar on and H0 is reduced by factors 3 and 6, respectively, compared to CMB only data, due to the measurement of matter clustering provided by the two observables. The cross-correlation signal has a constraining power that is comparable to the autocorrelation one and combining all the clustering measurements a sub-per cent error bar of 0.33 per cent on H0 can be achieved, which is about a factor 2 better than CMB only measurements. Finally, as a proof of concept, we test the full pipeline on the real data measured by the MeerKat collaboration (Cunnington et al. 2022) presenting some (weak) constraints on cosmological parameters.
