We consider Standard Model Higgs boson production in association with a W boson in hadron collisions. We supplement the fully exclusive perturbative computation of QCD radiative effects up to next-to-next-to-leading order (NNLO) with the computation of the decay of the Higgs boson into a pair at next-to-leading order (NLO). We consider the selection cuts that are typically applied in the LHC experimental analysis, and we compare our fixed-order predictions with the results obtained with the MC@NLO event generator. We find that NLO corrections to the H → decay can be important to obtain a reliable p T spectrum of the Higgs boson, but that, in the cases of interest, their effect is well accounted for by the parton shower Monte Carlo. NNLO corrections to the production process typically decrease the cross section by an amount which depends on the detail of the applied cuts, but they have a mild effect on the shape of the Higgs p T spectrum. We also discuss the effect of QCD radiative corrections on the invariant mass distribution of the Higgs candidate.