This article presents a comprehensive investigation of Neutral Fractional Delay Differential Equations (NFDDEs) with variable coefficients. The model incorporates Caputo fractional derivatives of orders \( \alpha \in (0,1] \) and \( \beta \in (0,\alpha] \), corresponding to the present and delayed state variables, respectively. The existence of solutions is established within an appropriate weighted-norm framework using the Banach Contraction Principle. In addition, Ulam-Hyers stability of the considered system is rigorously analyzed. From a computational perspective, an Euler-Fractional Wavelets-based algorithm is developed to obtain approximate solutions of the NFDDEs. A detailed error analysis of the proposed numerical scheme is carried out, demonstrating a computational accuracy of order \(O\!\left(h^{3-\alpha}\right)\). Several illustrative examples are presented to validate the theoretical results and to highlight the efficiency and robustness of the proposed method for different choices of the fractional orders \(\alpha\) and \(\beta\).