Conducting Uncertainty Quantification (UQ) and Global Sensitivity Analysis (GSA) on computationally expensive hydrocode simulations is challenging with sample-based methods. For this reason, the Hypercomplex Automatic Differentiation UQ (HYPAD-UQ) method was implemented in a Eulerian hydrocode. First- and higher-order partial derivatives of all code outputs with respect to each random variable are computed in a single run using HYPAD. These derivatives are used to construct surrogate models via the Taylor series expansion at each degree of freedom and at each time step. The expectation, variance, Sobol' indices, Shapley effects and sensitivities of these metrics with respect to distribution parameters of the random variables are then approximated from the Taylor series. HYPAD-UQ is significantly more computational efficient compared to sampling-based surrogate models like Gaussian Process. The HYPAD-UQ method is demonstrated on several numerical examples.