Galaxies occupy different regions of the [O iii]λ5007/H β-versus-[N ii]λ6584/H α emission-line ratio diagram in the distant and local Universe. We investigate the origin of this intriguing result by modelling self-consistently, for the first time, nebular emission from young stars, accreting black holes (BHs) and older, post-asymptotic giant branch (post-AGB) stellar populations in galaxy formation simulations in a full cosmological context. In post-processing, we couple new-generation nebular-emission models with high-resolution, cosmological zoom-in simulations of massive galaxies to explore which galaxy physical properties drive the redshift evolution of the optical-line ratios [O iii]λ5007/H β, [N ii]λ6584/H α, [S ii]λλ6717, 6731/H α and [O i]λ6300/H α. The line ratios of simulated galaxies agree well with observations of both star-forming and active local Sloan Digital Sky Survey galaxies. Towards higher redshifts, at fixed galaxy stellar mass, the average [O iii]/H β is predicted to increase and [N ii]/H α, [S ii]/H α and [O i]/H α to decrease – widely consistent with observations. At fixed stellar mass, we identify star formation history, which controls nebular emission from young stars via the ionization parameter, as the primary driver of the cosmic evolution of [O iii]/H β and [N ii]/H α. For [S ii]/H α and [O i]/H α, this applies only to redshifts greater than z = 1.5, the evolution at lower redshift being driven in roughly equal parts by nebular emission from active galactic nuclei and post-AGB stellar populations. Instead, changes in the hardness of ionizing radiation, ionized-gas density, the prevalence of BH accretion relative to star formation and the dust-to-metal mass ratio (whose impact on the gas-phase N/O ratio we model at fixed O/H) play at most a minor role in the cosmic evolution of simulated galaxy line ratios.