  Theoretical uncertainties on non-linear scales are among the main obstacles
to exploit the sensitivity of forthcoming galaxy and hydrogen surveys like
Euclid or the Square Kilometre Array (SKA). Here, we devise a new method to
model the theoretical error that goes beyond the usual cut-off on small scales.
The advantage of this more efficient implementation of the non-linear
uncertainties is tested through a Markov-Chain-Monte-Carlo (MCMC) forecast of
the sensitivity of Euclid and SKA to the parameters of the standard
$\Lambda$CDM model, including massive neutrinos with total mass $M_\nu$, and to
3 extended scenarios, including 1) additional relativistic degrees of freedom
($\Lambda$CDM + $M_\nu$ + $N_\mathrm{eff}$), 2) a deviation from the
cosmological constant ($\Lambda$CDM + $M_\nu$ + $w_0$), and 3) a time-varying
dark energy equation of state parameter ($\Lambda$CDM + $M_\nu$ +
$\left(w_0,w_a \right)$). We compare the sensitivity of 14 different
combinations of cosmological probes and experimental configurations. For Euclid
combined with Planck, assuming a plain cosmological constant, our method gives
robust predictions for a high sensitivity to the primordial spectral index
$n_{\rm s}$ ($\sigma(n_s)=0.00085$), the Hubble constant $H_0$
($\sigma(H_0)=0.141 \, {\rm km/s/Mpc}$), the total neutrino mass $M_\nu$
($\sigma(M_\nu)=0.020 \, {\rm eV}$). Assuming dynamical dark energy we get
$\sigma(M_\nu)=0.030 \, {\rm eV}$ for the mass and $(\sigma(w_0), \sigma(w_a))
= (0.0214, 0.071)$ for the equation of state parameters. The predicted
sensitivity to $M_\nu$ is mostly stable against the extensions of the
cosmological model considered here. Interestingly, a significant improvement of
the constraints on the extended model parameters is also obtained when
combining Euclid with a low redshift HI intensity mapping survey by SKA1,
demonstrating the importance of the synergy of Euclid and SKA.
