We compute synthetic, rest-frame optical and ultraviolet (UV) emission-line properties of galaxy populations at redshifts from z approximate to 0 to =8 in a full cosmological framework. We achieve this by coupling, in post-processing, the cosmological IllustrisTNG simulations with new-generation nebular-emission models, accounting for line emission from young stars, post-asymptotic giant branch (PAGB) stars, accreting black holes (BHs) and, for the first time, fast radiative shocks. The optical emission-line properties of simulated galaxies dominated by different ionizing sources in our models are largely consistent with those expected from classical diagnostic diagrams and reflect the observed increase in [Oiii]/H beta at fixed [Nii]/H alpha and the evolution of the H alpha, [Oiii]lambda 5007, and [Oii]lambda 3727 luminosity functions from z approximate to 0 to similar to 2. At higher redshift, we find that the emission-line galaxy population is dominated by star-forming and active galaxies, with negligible fractions of shock- and PAGB-dominated galaxies. We highlight 10 UV-diagnostic diagrams able to robustly identify the dominant ionizing sources in high-redshift galaxies. We also compute the evolution of several optical- and UV-line-luminosity functions from z=4 to 7, and the number of galaxies expected to be detectable per field of view in deep, medium-resolution spectroscopic observations with the NIRSpec instrument on board the JWST. We find that 2-h-long exposures are sufficient to achieve unbiased censuses of H alpha and [Oiii]lambda 5007 emitters, while at least 5 h are required for H beta, and even 10 h will detect only progressively smaller fractions of [Oii]lambda 3727, Oiii]lambda 1663, Ciii]lambda 1908, Civ lambda 1550, [Nii]lambda 6584, Siiii]lambda 1888, and Heii lambda 1640 emitters, especially in the presence of dust.