A search is presented for new particles in an extension to the Standard Model that includes a heavy Higgs boson (H0), an intermediate charged Higgs-boson pair (H±), and a light Higgs boson (h0). The analysis searches for events involving the production of a single heavy neutral Higgs boson which decays to the charged Higgs boson and a W boson, where the charged Higgs boson subsequently decays into a W boson and the lightest neutral Higgs boson decaying to a bottom–antibottom-quark pair. Such a cascade results in a W-boson pair and a bottom–antibottom-quark pair in the final state. Events with exactly one lepton, missing transverse momentum, and at least four jets are selected from a data sample corresponding to an integrated luminosity of 20.3  fb−1, collected by the ATLAS detector in proton-proton collisions at s=8  TeV at the LHC. The data are found to be consistent with Standard Model predictions, and 95% confidence-level upper limits are set on the product of cross section and branching ratio. These limits range from 0.065 to 43 pb as a function of H0 and H± masses, with mh0 fixed at 125 GeV.