This paper proposes a novel modified one-step iterative scheme for approximating common fixed points of three nonexpansive mappings in uniformly convex Banach spaces. The proposed scheme extends the classical one-step iteration associated with two mappings, which can be recovered as a particular case by an appropriate choice of the third mapping. Weak convergence of the generated sequence is established under both the Kadec-Klee property (KKP) and the Opial condition, while strong convergence is obtained by assuming a modified condition (B). Moreover, we introduce a new Condition (P) tailored for three mappings, which reduces to the well-known Condition (S) in the case of two mappings. The results presented here broaden and unify several existing contributions in the theory of common fixed point approximation.