This article presents a search for new resonances decaying into a Z or W boson and a 125 GeV Higgs boson h, and it targets the νν¯ bb¯ , ℓ⁺ℓ⁻bb¯ , or ℓ^±νbb¯ final states, where ℓ = e or μ, in proton-proton collisions at s = 13 TeV. The data used correspond to a total integrated luminosity of 139 fb⁻¹ collected by the ATLAS detector during Run 2 of the LHC at CERN. The search is conducted by examining the reconstructed invariant or transverse mass distributions of Zh or Wh candidates for evidence of a localised excess in the mass range from 220 GeV to 5 TeV. No significant excess is observed and 95% confidence-level upper limits between 1.3 pb and 0.3 fb are placed on the production cross section times branching fraction of neutral and charged spin-1 resonances and CP-odd scalar bosons. These limits are converted into constraints on the parameter space of the Heavy Vector Triplet model and the two-Higgs-doublet model. [Figure not available: see fulltext.].