We perform an error analysis of a fully discretised Streamline Upwind Petrov Galerkin Dynamical Low Rank (SUPG-DLR) method for random time-dependent advection-dominated problems. The time integration scheme has a splitting-like nature, allowing for potentially efficient computations of the factors characterising the discretised random field. The method allows to efficiently compute a low-rank approximation of the true solution, while naturally "inbuilding" the SUPG stabilisation. Standard error rates in the L2 and SUPG-norms are recovered. Numerical experiments validate the predicted rates.