An important aspect of solving the long-standing question as to what triggers various types of active galactic nuclei (AGN) involves a thorough understanding of the overall properties and formation history of their host galaxies. This is the second in a series of papers that systematically study the large-scale properties of cold neutral hydrogen (H i) gas in nearby radio galaxies. The main goal is to investigate the importance of gas-rich galaxy mergers and interactions among radio-loud AGN. In this paper, we present results of a complete sample of classical low-power radio galaxies. We find that extended Fanaroff & Riley type-I radio sources are generally not associated with gas-rich galaxy mergers or ongoing violent interactions, but occur in early-type galaxies without large (>rsim 108M⊙) amounts of extended neutral hydrogen gas. In contrast, enormous discs/rings of H i gas (with sizes up to 190 kpc and masses up to 2 × 1010 M⊙) are detected around the host galaxies of a significant fraction of the compact radio sources in our sample. This segregation in H i mass with radio-source size likely indicates that either these compact radio sources are confined by large amounts of gas in the central region or that their fuelling is inefficient and different from the fuelling process of classical FR I radio sources. To first order, the overall H i properties of our complete sample (detection rate, mass and morphology) appear similar to those of radio-quiet early-type galaxies. If confirmed by better statistics, this would imply that low-power radio-AGN activity may be a short and recurrent phase that occurs at some point during the lifetime of many early-type galaxies.