In the present work we study Bayesian nonparametric inference for the continuous-time M/G/1 queueing system. In the focus of the study is the unobservable service time distribution. We assume that the only available data of the system are the marked departure process of customers with the marks being the queue lengths just after departure instants. These marks constitute an embedded Markov chain whose distribution may be parametrized by stochastic matrices of a special delta form. We develop the theory in order to obtain integral mixtures of Markov measures with respect to suitable prior distributions. We have found a sufficient statistic with a distribution of a so-called S-structure sheding some new light on the inner statistical structure of the M/G/1 queue. Moreover, it allows to update suitable prior distributions to the posterior. Our inference methods are validated by large sample results as posterior consistency and posterior normality.