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 $M_ u$, and to 3 extended scenarios, including 1) additional relativistic degrees of freedom ($Lambda$CDM + $M_ u$ + $N_mathrm{eff}$), 2) a deviation from the cosmological constant ($Lambda$CDM + $M_ u$ + $w_0$), and 3) a time-varying dark energy equation of state parameter ($Lambda$CDM + $M_ u$ + $left(w_0,w_a right)$). 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 $M_ u$ ($sigma(M_ u)=0.020 , {rm eV}$). Assuming dynamical dark energy we get $sigma(M_ u)=0.030 , {rm eV}$ for the mass and $(sigma(w_0), sigma(w_a)) = (0.0214, 0.071)$ for the equation of state parameters. The predicted sensitivity to $M_ u$ 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.
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