Magnetic reconnection is an energy release mechanism fundamental to both astrophysical and laboratory plasmas, lying at the heart of phenomena including solar and stellar flares, geomagnetic substorms and tokamak disruptions in fusion plasmas. Here we detail the dynamic evolution of localised reconnection regions about three-dimensional (3D) magnetic null points by using numerical simulation. We demonstrate that reconnection triggered by the localised collapse of a 3D null point due to an external MHD wave involves a self-generated oscillation, whereby the current sheet and outflow jets undergo a reconnection reversal process during which back-pressure formation at the jet heads acts to prise open the collapsed field before overshooting the equilibrium into an opposite-polarity configuration. The discovery that reconnection at fully 3D nulls can proceed naturally in a time-dependent and periodic fashion is suggestive that oscillatory reconnection mechanisms may play a role in explaining periodicity in astrophysical phenomena associated with magnetic reconnection, such as the observed quasi-periodicity of solar and stellar flare emission. Furthermore, we find a consequence of oscillatory reconnection is the generation of a plethora of freely-propagating MHD waves which escape the vicinity of the reconnection region.