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 root 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 upper limits for sigma x B( A -> ZH) x B(H -> bb or H -> WW) 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 -> WW channel. An interpretation of the results in the context of two-Higgs-doublet models is also given.