We investigate the nebular emission produced by young stellar populations using the new GALSEVN model based on the combination of the SEVN population-synthesis code including binary-star processes and the GALAXEV code for the spectral evolution of stellar populations. Photoionization calculations performed with the CLOUDY code confirm that accounting for binary-star processes strongly influences the predicted emission-line properties of young galaxies. In particular, we find that our model naturally reproduces the strong HeII/Hb ratios commonly observed at high Hb equivalent widths in metal-poor, actively star-forming galaxies, which have proven challenging to reproduce using previous models. Including bursty star formation histories broadens the agreement with observations, while the most extreme HeII equivalent widths can be reproduced by models dominated by massive stars. GALSEVN also enables us to compute, for the first time in a way physically consistent with stellar emission, the emission from accretion discs of X-ray binaries (XRBs) and radiative shocks driven by stellar winds and supernova explosions. We find that these contributions are unlikely to prominently affect the predicted HeII/Hb ratio, and that previous claims of a significant contribution by XRBs to the luminosities of high-ionization lines are based on models predicting improbably high ratios of X-ray luminosity to star formation rate, inconsistent with the observed average luminosity function of XRBs in nearby galaxies. The results presented here provide a solid basis for a more comprehensive investigation of the physical properties of observed galaxies with GALSEVN using Bayesian inference.