Altermagnets (AMs) exhibit momentum-dependent spin splitting without net magnetization, making them promising platforms for spintronic applications. While the symmetry and multipolar origins of nonrelativistic altermagnetic spin splitting (NRASS) are well established, the role of electronic correlations and spin-lattice coupling (SLC) in controlling NRASS remain unexplored. In particular, SLC provides a direct pathway for tuning altermagnetism through lattice degrees of freedom. Here we first investigate the role of electronic correlations on altermagnetic strength in NiAs-type material CrSb using density functional theory (DFT), DFT+U, and dynamical mean-field theory (DMFT). We find that increasing electronic correlations substantially enhance NRASS and drive the stabilization of an incommensurate spin-spiral (SS) state, while dynamical correlations further amplify the spin splitting through quasiparticle renormalization of the Cr-3d states. We find that the NRASS remains 12.5% larger in DFT+DMFT than in DFT. By evaluating SLC in next step, we establish its direct connection to the evolution of NRASS and identify SLC as a microscopic descriptor between electronic correlation and enhanced altermagnetism in CrSb. Investigating it in another NiAs-type compound MnTe, we show that the SLC-NRASS correlation is generic across AMs and establish SLC as a microscopic descriptor of altermagnetic strength. Our results provide a unified framework for tuning altermagnetism through the interplay between the electronic correlations and lattice degrees of freedom.