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This article introduces an hp‐adaptive multi‐element stochastic collocation method, which additionally allows to re‐use existing model evaluations during either h‐ or p‐refinement. The collocation method is based on weighted Leja nodes. After h‐refinement, local interpolations are stabilized by adding and sorting Leja nodes on each newly created sub‐element in a hierarchical manner. For p‐refinement, the local polynomial approximations are based on total‐degree or dimension‐adaptive bases. The method is applied in the context of forward and inverse uncertainty quantification to handle non‐smooth or strongly localized response surfaces. The performance of the proposed method is assessed in several test cases, also in comparison to competing methods.