Measurements of the differential production cross-sections of prompt and non-prompt $J/\psi $ and $\psi (2{\textrm{S}})$ mesons with transverse momenta between 8 and 360 GeV and rapidity in the range $|y|<2$ are reported. Furthermore, measurements of the non-prompt fractions of $J/\psi $ and $\psi (2{\textrm{S}})$, and the prompt and non-prompt $\psi (2{\textrm{S}})$-to-$J/\psi $ production ratios, are presented. The analysis is performed using 140 fb$^{-1}$ of $\sqrt{s}=13$ TeV pp collision data recorded by the ATLAS detector at the LHC during the years 2015–2018.