To cope with load disturbance often encountered when performing identification tests on non-linear systems with input delay in industrial applications, a bias-eliminated Hammerstein-type output error (OE) model identification method is proposed in this study. By taking into account the load disturbance response as a time-varying parameter for estimation, two recursive least-squares (RLS) identification algorithms is established to estimate the Hammerstein-type model parameters and the time-varying disturbance response. A one-dimensional searching approach is adopted to determine the integer-type delay parameter by minimising the fitting error of output response. Moreover, an auxiliary OE model is constructed to ensure consistent estimation under stochastic noise. In addition, two adaptive forgetting factors are introduced into the proposed RLS algorithms to enhance the estimation convergence on the model parameters and the disturbance response. Asymptotic properties of parameter estimation are analysed along with a proof, in particular for unbiased estimation against a constant disturbance. Two illustrative examples are given to demonstrate the effectiveness of the proposed identification method.