Despite their compensated magnetic moments, altermagnets (AMs) exhibit nonrelativistic spin splitting (NRSS) that offers routes to spintronic functionality without relying on relativistic spin-orbit coupling. However, how NRSS influences relativistic phenomena remains largely unexplored. Here we show that NRSS plays a crucial role in reshaping the Rashba effect in AMs with broken inversion symmetry. Using a tight-binding model, we demonstrate that Rashba spin splitting, whose magnitude is typically limited by the strength of spin-orbit coupling, is governed by NRSS energy scales. The resulting Rashba bands combined with NRSS generate anomalously large charge-to-spin conversion with a Néel-vector-tunable spin polarization. First-principles calculations for the noncentrosymmetric AM GdAlSi corroborate the emergence of these effects in a real system. Our results reveal the interplay between nonrelativistic and relativistic effects in AMs and identify them as fertile platforms for spin-orbitronic applications.