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$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ 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 upper limits for $\sigma \times B(A\to ZH)\times B(H\to bb$ or $H\to WW)$ are set. The upper limits are in the range 0.0062-0.380 pb for the $H\to bb$ channel and in the range 0.023-8.9 pb for the $H\to WW$ channel. An interpretation of the results in the context of two-Higgs-Doublet models is also given.