This paper deals with the introduction of a multiresolution strategy into the semi-intrusive scheme, recently introduced by the authors, aiming to propagate uncertainties in unsteady compressible fluid applications. The mathematical framework of the multiresolution setting is presented for the cell-average case, and the coupling with the semi-intrusive scheme is described from both the theoretical and algorithmic point-of-view. Some reference test cases are performed to demonstrate the convergence properties and the efficiency of the overall scheme: the linear advection problem for both smooth and discontinuous initial conditions, the inviscid Burgers equation, and an uncertain shock tube problem obtained by modifying the well-known Sod shock problem. For all the cases, the convergence curves are computed with respect to semi-analytical (exact) solutions. In the case of the shock tube problem, an original technique to obtain a reference highly-accurate numerical stochastic solution has also been developed.