Data-driven predictive control promises model-free wave-dampening strategies for Connected and Autonomous Vehicles (CAVs) in mixed traffic flow. However, its performance relies on data quality, which suffers from unknown noise and disturbances. This paper introduces a Robust Data-EnablEd Predictive Leading Cruise Control (RDeeP-LCC) method based on reachability analysis, aiming to achieve safe and optimal CAV control under bounded process noise and external disturbances. Precisely, the matrix zonotope set technique and Willems' Fundamental Lemma are employed to derive the over-approximated system dynamics directly from data, and a data-driven feedback control technique is utilized to obtain an additional feedback input for stability. We decouple the mixed platoon into an error system and a nominal system, where the error system provides data-driven reachability sets for the enhanced safety constraints in the nominal system. Finally, a data-driven predictive control framework is formulated in a tube-based control manner for robustness guarantees. Nonlinear simulations with noise-corrupted data demonstrate that the proposed method outperforms baseline methods in mitigating traffic waves.