In this work, we propose a semi-analytical approach based on the Sumudu–Adomian Decomposition Method (SADM) for nonlinear fractional population models governed by the Caputo–Fabrizio derivative. The proposed approach integrates the Adomian decomposition method with the Sumudu transform. The solution of the nonlinear fractional initial-value problem is represented by a rapidly convergent decomposition series. Existence and uniqueness of the solution are demonstrated under Lipschitz-type conditions. The convergence of the derived decomposition series, an explicit truncation-error estimate, and uniform stability are established. Moreover, model-specific verification of the sufficient contraction condition is provided over appropriate biologically admissible intervals to link the theoretical analysis with the numerical applications. The method is applied to four representative population-growth models: the logistic model, Allee-effect model, Richards model and Gompertz model. Comparison of the SADM approximations with the corresponding numerical reference solutions confirms the reliability and accuracy of the proposed method.