A search for charged Higgs bosons decaying into W$^{±}$W$^{±}$ or W$^{±}$Z bosons is performed, involving experimental signatures with two leptons of the same charge, or three or four leptons with a variety of charge combinations, missing transverse momentum and jets. A data sample of proton-proton collisions at a centre-of-mass energy of 13 TeV recorded with the ATLAS detector at the Large Hadron Collider between 2015 and 2018 is used. The data correspond to a total integrated luminosity of 139 fb$^{−1}$. The search is guided by a type-II seesaw model that extends the scalar sector of the Standard Model with a scalar triplet, leading to a phenomenology that includes doubly and singly charged Higgs bosons. Two scenarios are explored, corresponding to the pair production of doubly charged H$^{±±}$ bosons, or the associated production of a doubly charged H$^{±±}$ boson and a singly charged H$^{±}$ boson. No significant deviations from the Standard Model predictions are observed. H$^{±±}$ bosons are excluded at 95% confidence level up to 350 GeV and 230 GeV for the pair and associated production modes, respectively.