Measurements of the total and differential Higgs boson production cross-sections, via WH and ZH associated production using H → WW$^{*}$ → ℓνℓν and H → WW$^{*}$ → ℓνjj decays, are presented. The analysis uses proton-proton events delivered by the Large Hadron Collider at a centre-of-mass energy of 13 TeV and recorded by the ATLAS detector between 2015 and 2018. The data correspond to an integrated luminosity of 140 fb$^{−1}$. The sum of the WH and ZH cross-sections times the H → WW$^{*}$ branching fraction is measured to be $ {0.44}_{-0.09}^{+0.10}{\left(\textrm{stat}.\right)}_{-0.05}^{+0.06}\left(\textrm{syst}.\right) $ pb, in agreement with the Standard Model prediction. Higgs boson production is further characterised through measurements of the differential cross-section as a function of the transverse momentum of the vector boson and in the framework of Simplified Template Cross-Sections.[graphic not available: see fulltext]