  Next-generation cosmological surveys will observe larger cosmic volumes than
ever before, enabling us to access information on the primordial Universe, as
well as on relativistic effects. We consider forthcoming 21cm intensity mapping
surveys (SKAO) and optical galaxy surveys (DESI and Euclid), combining the
information via multi-tracer cross-correlations that suppress cosmic variance
on ultra-large scales. In order to fully incorporate wide-angle effects and
redshift-bin cross-correlations, together with lensing magnification and other
relativistic effects, we use the angular power spectra, $C_\ell(z_i,z_j)$.
Applying a Fisher analysis, we forecast the expected precision on $f_{\rm NL}$
and the detectability of lensing and other relativistic effects. We find that
the full combination of two pairs of 21cm and galaxy surveys, one pair at low
redshift and one at high redshift, could deliver $\sigma(f_{\rm NL})\sim 1.5$,
detect the Doppler effect with a signal-to-noise ratio $\sim$8 and measure the
lensing convergence contribution at $\sim$2\% precision. In a companion paper,
we show that the best-fit values of $f_{\rm NL}$ and of standard cosmological
parameters are significantly biased if the lensing contribution neglected.
