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 $\sqrt{s} = 13$ $\text {TeV}$ 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 $\text {GeV}$ for the A boson and 130–700 $\text {GeV}$ 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 $\sigma \times B(A\rightarrow ZH)\times B(H\rightarrow bb \; \text {or} \; H \rightarrow WW)$ are set. The upper limits are in the range 0.0062–0.380 pb for the $H\rightarrow bb$ channel and in the range 0.023–8.9 pb for the $H\rightarrow WW$ channel. An interpretation of the results in the context of two-Higgs-doublet models is also given.