We present the results of a high-resolution, 5GHz, Karl G. Jansky Very Large Array study of the nuclear radio emission in a representative subset of the atlas^3D^ survey of early-type galaxies (ETGs). We find that 51+/-4 per cent of the ETGs in our sample contain nuclear radio emission with luminosities as low as 10^18^W/Hz. Most of the nuclear radio sources have compact (<~25-110pc) morphologies, although ~10 per cent display multicomponent core+jet or extended jet/lobe structures. Based on the radio continuum properties, as well as optical emission line diagnostics and the nuclear X-ray properties, we conclude that the majority of the central 5GHz sources detected in the atlas^3D^ galaxies are associated with the presence of an active galactic nucleus (AGN). However, even at subarcsecond spatial resolution, the nuclear radio emission in some cases appears to arise from low-level nuclear star formation rather than an AGN, particularly when molecular gas and a young central stellar population is present. This is in contrast to popular assumptions in the literature that the presence of a compact, unresolved, nuclear radio continuum source universally signifies the presence of an AGN. Additionally, we examine the relationships between the 5 GHz luminosity and various galaxy properties including the molecular gas mass and - for the first time - the global kinematic state. We discuss implications for the growth, triggering, and fuelling of radio AGNs, as well as AGN-driven feedback in the continued evolution of nearby ETGs.