We present the results of a comparative ab initio study of single-walled SiC, BN, and BeO nanotubes (NTs) in zigzag and armchair configurations. Within density functional theory, we employ self-interaction-corrected pseudopotentials that were shown previously to yield reliable results for both structural and electronic properties of related bulk crystals. Using these pseudopotentials, we investigate the dependence of the atomic relaxation, strain energy, Young's modulus, and electronic structure on nanotube diameter and compound ionicity. Qualitatively, the NTs of all three wide-band-gap compounds show similar radially buckled geometries upon atomic relaxation, similar strain energy progressions with NT diameter and a saturation of Young's modulus as well as the band gap energy for large NT diameters. The band gap progression with NT diameter, which is of crucial importance for device applications, is presented and analyzed in detail. For SiC and BN, the calculated band gap energies of zigzag NTs vary much stronger for small and medium diameters than those of their armchair counterparts showing a significant narrowing of the band gaps. In contrast, the band gap progression in zigzag and armchair BeO NTs shows a very peculiar behavior for small diameters. No band gap breakdown occurs and the gap goes through a minimum for zigzag BeO NTs. The qualitative difference in the nature of the lower conduction band states in SiC and BN NTs, as compared to BeO NTs, and the increasing ionicity of these compounds are shown to be responsible for the observed effects.