Offshore wind turbines intend to take a rapidly growing share in the electric mix. The design, installation, and exploitation of these industrial assets are regulated by international standards, providing generic guidelines. Constantly, new projects reach unexploited wind resources, pushing back installation limits. Therefore, turbines are more and more subject to uncertain environmental conditions, making long-term investment decisions riskier. Fortunately, numerical models of wind turbines can perform accurate multi-physics simulations of such systems interacting with their environment. The challenge is then to propagate the input environmental uncertainties through such a model and analyze the distribution of model outputs of interest. Since each simulation of this numerical model is costly, the estimation of output quantities of interest (e.g., mean, variance) must be done with a restricted number of simulations. To do so, this paper combines kernel herding sampling with Bayesian quadrature to estimate the mean fatigue damage,with an application to an offshore wind turbine operating in Teesside, UK. It is demonstrated theoretically and numerically that this method guarantees fast and accurate convergence. Moreover, this approach is highly flexible by allowing to subsample directly from a given dataset and being fully distributable on high-performance computing facilities. Finally, a new Python package was developed and documented to provide quick open access to this uncertainty propagation method.