This paper considers stochastic simulations with correlated input random variables having NORmal-To-Anything (NORTA) distributions. We assume that the simulation analyst does not know the marginal distribution functions and the base correlation matrix of the NORTA distribution but has access to a finite amount of input data for statistical inference. We propose a Bayesian procedure that decouples the input model estimation into two stages and overcomes the problem of inconsistently estimating the base correlation matrix of the NORTA distribution in the presence of discrete input variables. It further allows us to estimate the variability of the simulation output data that are attributable to the input uncertainty due to not knowing the NORTA distribution. Using this input uncertainty estimate, we introduce a simple yet effective method to obtain input uncertainty adjusted credible intervals. We illustrate our method in an assemble-to-order production system with a correlated demand arrival process.