A search for charged Higgs bosons decaying into $W\pm W\pm$ or $W\pm Z$ 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$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$. 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 $H\pm \pm$ bosons, or the associated production of a doubly charged $H\pm \pm$ boson and a singly charged $H\pm$ boson. No significant deviations from the Standard Model predictions are observed. $H\pm \pm$ bosons are excluded at 95% confidence level up to 350 GeV and 230 GeV for the pair and associated production modes, respectively.