Context. An injection of energy towards a magnetic null point can drive reversals of current-sheet polarity leading to time-dependent, oscillatory reconnection (OR), which may explain periodic phenomena generated when reconnection occurs in the solar atmosphere. However, the details of what controls the period of these current-sheet oscillations in realistic systems is poorly understood, despite being of crucial importance in assessing whether a specific model of OR can account for observed periodic behaviour. Aims. This paper aims to highlight that different types of reconnection reversal are supported about null points, and that these can be distinct from the oscillation in the closed-boundary, linear systems considered by a number of authors in the 1990s. In particular, we explore the features of a nonlinear oscillation local to the null point, and examine the effect of resistivity and perturbation energy on the period, contrasting it to the linear, closed-boundary case. Methods. Numerical simulations of the single-fluid, resistive MHD equations are used to investigate the effects of plasma resistivity and perturbation energy upon the resulting OR. Results. It is found that for small perturbations that behave linearly, the inverse Lundquist number dictates the period, provided the perturbation energy (i.e. the free energy) is small relative to the inverse Lundquist number defined on the boundary, regardless of the broadband structure of the initial perturbation. However, when the perturbation energy exceeds the threshold required for "nonlinear" null collapse to occur, a complex oscillation of the magnetic field is produced which is, at most, only weakly-dependent on the resistivity. The resultant periodicity is instead strongly influenced by the amount of free energy, with more energetic perturbations producing higher-frequency oscillations. Conclusions. Crucially, with regards to typical solar-based and astrophysical-based input energies, we demonstrate that the majority far exceed the threshold for nonlinearity to develop. This substantially alters the properties and periodicity of both null collapse and subsequent OR. Therefore, nonlinear regimes of OR should be considered in solar and astrophysical contexts.