A search for heavy right-handed Majorana or Dirac neutrinos N-R and heavy right-handed gauge bosons W-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 root 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-R boson and N-R plane. The excluded region extends to about m(W-R) = 6.4 TeV for both Majorana and Dirac N-R neutrinos at m(N-R) < 1 TeV. N-R with masses of less than 3.5 (3.6) TeV are excluded in the electron (muon) channel at m(W-R) = 4.8 TeV for the Majorana neutrinos, and limits of m(N-R) up to 3.6 TeV for m(W-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.