[en] The ATLAS collaboration recently reported on the first observation of associated-production of a $Z^0$ boson with a $J/\psi$. We recently claimed that the corresponding yield of the prompt $J/\psi$ was dominated by double parton scatterings in the ATLAS acceptance with a somewhat small value of ${\sigma }_{\mathrm{eff}}$. We also found out that single parton scatterings were only dominant at large transverse momenta. We present here the first phenomenological analysis of another part of the ATLAS data sample, namely of a $Z^0$ boson plus a non-prompt $J/\psi$. Our study is performed at next-to-leading order in ${\alpha }_s$ and includes parton-shower effects via the MadGraph5_[email protected] framework. We find out that the data, unlike the case of prompt $J/\psi +Z^0$, do not hint at significant DPS contributions. Owing to the current experimental and theoretical uncertainties, there is still a room for these but with a lower limit of ${\sigma }_{\mathrm{eff}}$ close to 5 mb. We stress the importance of QCD corrections to account for the ATLAS data.