Polynomial chaos expansions (PCE) have proven efficiency in a number of fields for propagating parametric uncertainties through computational models of complex systems, namely structural and fluid mechanics, chemical reactions and electromagnetism, etc. For problems involving oscillatory, time-dependent output quantities of interest, it is well-known that reasonable accuracy of PCE-based approaches is difficult to reach in the long term. In this paper, we propose a fully non-intrusive approach based on stochastic time warping to address this issue: each realization (trajectory) of the model response is first rescaled to its own time scale so as to put all sampled trajectories in phase in a common virtual timeline. Principal component analysis is introduced to compress the information contained in these transformed trajectories and sparse PCE representations using least angle regression are finally used to approximate the components. The approach shows remarkably small prediction error for particular trajectories as well as for second-order statistics of the latter. It is illustrated on different benchmark problems well known in the literature on time-dependent PCE problems, ranging from rigid body dynamics, chemical reactions to forced oscillations of a non-linear system.