A combination of searches for singly and doubly charged Higgs bosons, H± and H±±, produced via vector-boson fusion is performed using 140 fb−1 of proton–proton collisions at a centre-of-mass energy of 13 TeV, collected with the ATLAS detector during Run 2 of the Large Hadron Collider. Searches targeting decays to massive vector bosons in leptonic final states (electrons or muons) are considered. New constraints are reported on the production cross-section times branching fraction for charged Higgs boson masses between 200 GeV and 3000 GeV. The results are interpreted in the context of the Georgi-Machacek model for which the most stringent constraints to date are set for the masses considered in the combination.