The central regions of disc galaxies hold clues to the processes that dominate their formation and evolution. The TIMER project has obtained high signal-to-noise and spatial resolution integral-field spectroscopy data of the inner few kpc of 21 nearby massive barred galaxies, allowing studies of the stellar kinematics with unprecedented spatial resolution. We confirm theoretical predictions of the effects of bars on stellar kinematics, and identify box/peanuts through kinematic signatures in mildly and moderately inclined galaxies, finding a lower limit to the fraction of massive barred galaxies with box/peanuts at ~62%. Further, we provide kinematic evidence of the connection between barlenses, box/peanuts and bars. We establish the presence of nuclear discs in 19 galaxies and show that their kinematics are characterised by near-circular orbits with low pressure support, and are consistent with the bar-driven secular evolution picture for their formation. In fact, we show that these nuclear discs have, in the region where they dominate, larger rotational support than the underlying main galaxy disc. We define a kinematic radius for the nuclear discs and show that it relates to bar radius, ellipticity and strength, and bar-to-total ratio. Comparing our results with photometric studies, we find that state-of-the-art galaxy image decompositions are able to discern nuclear discs from classical bulges, if the images employed have enough physical spatial resolution. In fact, we show that nuclear discs are typically identified in such image decompositions as photometric bulges with (near-)exponential profiles. However, we find that the presence of composite bulges (galaxies hosting both a classical bulge and a nuclear disc) can often be unnoticed in studies based on photometry alone, and suggest a more stringent threshold to the Sersic index to identify galaxies with pure classical bulges.