A search for heavy right-handed Majorana or Dirac neutrinos $N_{\textrm{R}}$ and heavy right-handed gauge bosons $W_{\textrm{R}}$ is performed in events with energetic electrons or muons, with the same or opposite electric charge, and energetic jets. The search is carried out separately for topologies of clearly separated final-state products (“resolved” channel) and topologies with boosted final states with hadronic and/or leptonic products partially overlapping and reconstructed as a large-radius jet (“boosted” channel). The events are selected from pp collision data at the LHC with an integrated luminosity of 139 fb$^{-1}$ collected by the ATLAS detector at $\sqrt{s} = 13$ TeV. No significant deviations from the Standard Model predictions are observed. The results are interpreted within the theoretical framework of a left-right symmetric model, and lower limits are set on masses in the heavy right-handed $W_{\textrm{R}}$ boson and $N_{\textrm{R}}$ plane. The excluded region extends to about $m(W_{\textrm{R}}) = 6.4$ TeV for both Majorana and Dirac $N_{\textrm{R}}$ neutrinos at $m(N_{\textrm{R}})<1$ TeV. $N_{\textrm{R}}$ with masses of less than 3.5 (3.6) TeV are excluded in the electron (muon) channel at $m(W_{\textrm{R}})=4.8$ TeV for the Majorana neutrinos, and limits of $m(N_{\textrm{R}})$ up to 3.6 TeV for $m(W_{\textrm{R}}) = 5.2$ (5.0) TeV in the electron (muon) channel are set for the Dirac neutrinos. These constitute the most stringent exclusion limits to date for the model considered.