The centres of disc galaxies host a variety of structures built via both internal and external processes. In this study, we constrain the formation and evolution of these central structures, in particular nuclear rings and nuclear discs, by deriving maps of mean stellar ages, metallicities and [$\alpha$/Fe] abundances. We use observations obtained with the MUSE integral-field spectrograph for the TIMER sample of 21 massive barred galaxies. Our results indicate that nuclear discs and nuclear rings are part of the same physical component, with nuclear rings constituting the outer edge of nuclear discs. All nuclear discs in the sample are clearly distinguished based on their stellar population properties. As expected in the picture of bar-driven secular evolution, nuclear discs are younger, more metal-rich, and show lower [$\alpha$/Fe] enhancements, as compared to their immediate surroundings. Moreover, nuclear discs exhibit well-defined radial gradients, with ages and metallicities decreasing, and [$\alpha$/Fe] abundances increasing with radius out to the nuclear ring. Often, these gradients show no breaks from the edge of the nuclear disc until the centre, suggesting that these structures extend to the very centres of the galaxies. We argue that continuous (stellar) nuclear discs may form from a series of bar-built (initially gas-rich) nuclear rings that grow in radius, as the bar evolves. In this picture, nuclear rings are simply the (often) star-forming outer edge of nuclear discs. Finally, by combining our results with those from a accompanying kinematic study, we do not find evidence for the presence of large, dispersion-dominated components in the centres of these galaxies. This could be a result of quiet merger histories, despite the large galaxy masses, or perhaps high angular momentum and strong feedback processes preventing the formation of these kinematically hot components.