This paper develops a rigorous hybrid analytical--numerical framework that combines the Double Sumudu Transform (DST) with a parameter-optimized multistage Adomian Decomposition Method (ADM) for the solution of nonlinear time-fractional partial differential equations (PDEs) subject to mixed nonlinear boundary conditions. In this approach, the integral-transform properties of the DST are employed to reduce the complexity of fractional operators, while the multistage ADM, enhanced by metaheuristic parameter tuning, improves the convergence rate and reduces computational overhead. A theoretical foundation is provided by establishing error bounds and stability criteria in suitable Banach spaces, with explicit assumptions on the nonlinear operators. The accuracy and efficiency of the method are validated through benchmark problems, including the time-fractional heat equation, fractional Klein--Gordon equation, and nonlinear reaction--diffusion models with Robin- and cubic-type boundary conditions. Numerical comparisons against classical ADM, recent transform-based schemes, and numerical methods such as RBF-FD and spectral techniques demonstrate improved convergence, lower root-mean-square errors, and competitive computational cost. The results confirm that the proposed hybrid framework constitutes a mathematically consistent and practically efficient tool for solving complex nonlinear fractional PDEs, and it provides a viable alternative to standard transform or decomposition methods.