The primary focus of this paper is to present and demonstrate an efficient approach for propagating mixed inputuncertainties, aleatory and epistemic, through a complex simulation code. In particular, the approach is applied totheuncertaintyquantificationofsurfaceheatfluxtothesphericalnonablatingheat-shieldofagenericreentryvehicledue to epistemic and aleatory uncertainties that may exist in various parameters used in the numerical solution ofhypersonic, viscous, laminar blunt-bodyflows with thermochemical nonequilibrium. Two main uncertainty sourceswere treated in the computationalfluid dynamics simulations: 1) aleatory uncertainty in the freestream velocity and2) epistemic uncertainty in the recombination efficiency for a partially catalytic wall boundary condition. Thesecond-order uncertainty quantification using a stochastic response surface obtained with point-collocationnonintrusive polynomial chaos was used for the propagation of mixed (aleatory and epistemic) uncertainties. Theuncertainty quantification approach was thoroughly tested on a stochastic model problem with mixed uncertaintiesfor the prediction of stagnation-point heat transfer with Fay-Riddell relation, which included the comparison withdirect Monte Carlo sampling results. In the stochastic computationalfluid dynamics problem, the uncertainty insurface heat transfer was obtained in terms of intervals at different probability levels at various locations, includingthestagnationpointandtheshoulderregion.Themixeduncertaintyresultswerecomparedwiththeresultsobtainedwith a purely aleatory uncertainty analysis to show the difference between two uncertainty quantificationapproaches. A global sensitivity analysis indicated that the velocity has a stronger contribution to the overalluncertainty in the stagnation-point heat transfer.