This paper is the second in a series in which we present intermediate-resolution, wide-wavelength coverage spectra for a complete sample of 14 compact radio sources, taken with the aim of investigating the impact of the nuclear activity on the circumnuclear interstellar medium (ISM) in the early stages of radio source evolution. In the first paper (Holt, Tadhunter & Morganti), we presented the kinematic results from the nuclear emission-line modelling and reported fast outflows in the circumnuclear gas. In this paper, we use the line fluxes to investigate the physical conditions and dominant ionization mechanisms of the emission-line gas. We find evidence for large electron densities and high reddening in the nuclear regions, particularly in the broader, blueshifted components. These results are consistent with the idea that the young, recently triggered radio sources still reside in dense and dusty cocoons deposited by the recent activity triggering event (merger/interaction). In addition, we find that the quiescent nuclear and extended narrow components are consistent with active galactic nucleus (AGN) photoionization, split between simple-slab AGN photoionization and mixed-medium photoionization models. For the nuclear broader and shifted components, the results are less clear. Whilst there are suggestions that the broader components may be closer to shock plus precursor models on the diagnostic diagrams, and that the electron temperatures and densities are high, we are unable to unambiguously distinguish the dominant ionization mechanism using the optical emission-line ratios. This is surprising given the strong evidence for jet-cloud interactions (broad emission lines, large outflow velocities and strong radio-optical alignments), which favours the idea that the warm gas has been accelerated in shocks driven by the radio lobes expanding through a dense cocoon of gas deposited during the triggering event.