A search is presented for new resonances decaying to a or boson and a 125~\GeV\ Higgs boson (h) in the , , and final states, where or , in collisions at ~\TeV. The data used corresponds to a total of 36.1~fb of integrated luminosity collected with the ATLAS detector at the Large Hadron Collider from the 2015 and 2016 data taking periods. The search is conducted by examining the reconstructed invariant or transverse mass distributions of and candidates for evidence of a localised excess. The results are interpreted in terms of constraints on the production cross-section times branching fraction of heavy and resonances in Heavy Vector Triplet models or the CP odd scalar boson in Two-Higgs-Doublet models over the mass range from 220 GeV up to 5~TeV. No significant excess is observed. Limits are placed at 95\% CL on the production cross-section times branching fraction and range between and ~pb depending on the model and mass of the resonance.