The goal of this work is to determine the effect of grid topology and flow solver discretization on the uncertainty quantification and sensitivity analysis of commonly used turbulence models in Reynolds-averaged Navier-Stokes codes due to uncertainty in the values of turbulence model coefficients as well as to rank the contribution of each coefficient to uncertainty in various output quantities of interest for transonic flow over an RAE 2822 airfoil. Three turbulence models were considered: Spalart-Allmaras, 2006 Wilcox k-ω, and Menter shear-stress transport. Several structured and unstructured grid topologies were employed in the analysis. The FUN3D code of NASA Langley Research Center and the BCFD code of the Boeing Company were used as the flow solvers. The uncertainty analysis employed stochastic expansions based on nonintrusive polynomial chaos for efficient uncertainty propagation. Sobol indices were used to rank the relative contributions of each coefficient to the total uncertainty in the output quantities of interest. The solutions obtained with all grids and computational-fluid-dynamics codes identified a common set of coefficients for each turbulence model that influence the output uncertainty significantly for transonic wall-bounded flows.