This work presents a unified pipeline for building accurate surrogate models for forecasting that account for parameter and state uncertainty and enable time-varying sensitivity analysis for models with time-dependent outputs. Many well-established approaches for forward uncertainty quantification and global sensitivity analysis primarily address scalar quantities of interest. At the same time, practical dynamical models, such as those in hydrology and environmental sciences, produce time-series outputs. Extending these methods to vector-valued outputs requires a structured computational pipeline. We first apply Bayesian filtering using a particle filter to sequentially improve knowledge of uncertain model parameters and internal states by assimilating observations as they become available. The resulting posterior parameter distribution is then propagated through the model using a Polynomial Chaos Expansion-based surrogate to generate probabilistic forecasts at future time steps. To improve numerical stability and efficiency during surrogate construction, transport maps are used to transform the posterior distribution into a standard Gaussian space. Finally, the temporal evolution of parameter influence is quantified using generalized Sobol sensitivity indices. Results show that the proposed pipeline facilitates faster model evaluation, enables deeper insights into model behavior under varying conditions, and provides a practical framework for real-time predictive tasks such as flood forecasting and drought assessment