The vertical (Lense-Thirring) precession of the innermost accretion flows has been discussed as a sensitive indicator of the rotational properties of neutron stars (NSs) and their equation of state because it vanishes for a non-rotating star. For this work we applied the Hartle-Thorne spacetimes to study the frequencies of the precession for both geodesic and non-geodesic (fluid) flows. We built on previous findings of the effect of the NS quadrupole moment, which revealed the importance of the interplay between the relativistic and classical precession. Because of this interplay, the widely used Lense-Thirring metric, linear in the NS angular momentum, is insufficient to calculate the behaviour of the precession frequency across an astrophysically relevant range of NS angular momentum values. We find that even for moderately oblate NSs, the dependences of the precession frequency on the NS angular momentum at radii within the innermost accretion region have maxima that occur at relatively low values of the NS angular momentum. We conclude that very different groups of accreting NSs - slow and fast rotators - can display the same precession frequencies. This may explain the lack of evidence for a correlation between the frequencies of the observed low-frequency quasiperiodic oscillations and the NS spin. In our work we provide a full general description of precession behaviour, and also examples that assume specific NS and quark star (MIT bag) equations of state. Our calculations are reproducible using the associated Wolfram Mathematica notebook.