Models of nuclear dark matter propose that the dark sector contains large composite states consisting of dark nucleons in analogy to Standard Model nuclei. We examine the direct detection phenomenology of a particular class of nuclear dark matter model at the current generation of tonne-scale liquid noble experiments, in particular DEAP-3600 and XENON1T. In our chosen nuclear dark matter scenario distinctive features arise in the recoil energy spectra due to the non-point-like nature of the composite dark matter state. We calculate the number of events required to distinguish these spectra from those of a standard point-like WIMP state with a decaying exponential recoil spectrum. In the most favourable regions of nuclear dark matter parameter space, we find that a few tens of events are needed to distinguish nuclear dark matter from WIMPs at the $3\,\sigma$ level in a single experiment. Given the total exposure time of DEAP-3600 and XENON1T we find that at best a $2\,\sigma$ distinction is possible by these experiments individually, while $3\,\sigma$ sensitivity is reached for a range of parameters by the combination of the two experiments. We show that future upgrades of these experiments have potential to distinguish a large range of nuclear dark matter models from that of a WIMP at greater than $3\,\sigma$.