  We present a comprehensive set of forecasts for the cross-correlation signal
between 21cm intensity mapping and galaxy redshift surveys. We focus on the
data sets that will be provided by the SKAO for the 21cm 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 parameterization 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 ($k<0.25 h/$Mpc). The forecasted constraints, obtained
with Monte Carlo Markov Chains techniques in a Bayesian framework, in terms of
the six base parameters of the standard $\Lambda$CDM model, are promising. The
predicted signal-to-noise ratio for the cross-correlation can reach $\sim 50$
for $z\sim 1$ and $k\sim 0.1 h/$ Mpc. When the cross-correlation signal is
combined with current Cosmic Microwave Background (CMB) data from Planck, the
error bar on $\Omega_{\rm c}\,h^2$ and $H_0$ is reduced by a factor 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 auto-correlation one and combining
all the clustering measurements a sub-percent error bar of 0.33% on $H_0$ can
be achieved, which is about a factor 2 better than CMB only measurement.
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.
