A search for a heavy neutral Higgs boson, A, decaying into a Z boson and another heavy Higgs boson, H, is performed using a data sample corresponding to an integrated luminosity of 139 fb−1 from proton—proton collisions at √s = 13 TeV recorded by the ATLAS detector at the LHC. The search considers the Z boson decaying into electrons or muons and the H boson into a pair of b-quarks or W bosons. The mass range considered is 230—800 GeV for the A boson and 130—700 GeV for the H boson. The data are in good agreement with the background predicted by the Standard Model, and therefore 95% confidence-level upperlimits for σ × B(A → Z H ) × B(H → bb or H → W W )are set. The upper limits are in the range 0.0062—0.380 pb for the H → bb channel and in the range 0.023—8.9 pb for the H → W W  channel. An interpretation of the results in the context of two-Higgs-doublet models is also given.