We present a multi-resolution scheme, based on piecewise polynomial approximations at the stochastic level, for the resolution of nonlinear hyperbolic problems subjected to parametric uncertainties. The numerical method rely on a Galerkin projection technique at the stochastic level, with a finite-volume discretization and a Roe solver (with entropy corrector) in space and time. A key issue in uncertain hyperbolic problem is the loss of smoothness of the solution with regard to the uncertain parameters, which calls for piecewise continuous approximations and multi-resolution schemes, together with adaptive strategies. However, discontinuities in the spatial and stochastic domains are well localized, requiring very different discretization efforts according to the local smoothness of the solution. As a result, classical discretization approaches based on the tensorization of stochastic and deterministic approximation spaces (bases) are inefficient and we propose a numerical procedure where the spatial discretization is fixed while the stochastic basis is locally adapted in space to fit the solution complexity. Examples of applications and efficiency / complexity assessment of the method will be shown.