In Debattista et al. (2015), we proposed that a kiloparsec-scale nuclear disc is responsible for the high-velocity secondary peak in the stellar line-of-sight velocity distributions (LOSVDs) seen at positive longitudes in the bulge by the Apache Point Observatory Galactic Evolution Experiment (APOGEE). Here, we make further qualitative but distinctive predictions of the kinematic properties of a nuclear disc, including for the LOSVDs at negative longitudes (which APOGEE-2 will observe) and examine the proper motions throughout the disc. Since a nuclear ring is also able to produce similar high-velocity LOSVD peaks, we present predictions for the proper motion signatures which distinguish between a nuclear disc and a nuclear ring. We also demonstrate that the stars in a nuclear disc, which would be on x2 orbits perpendicular to the bar, can remain on these orbits for a long time and can therefore be old. We show that such (old) nuclear discs of comparable size exist in external galaxies.