CLASSY XIV: The Nitrogen Exception -- Multi-Phase Enrichment and Feedback in High-$z$ Analogs

We present a first-of-its-kind analysis of the metal content across two interstellar medium (ISM) phases in a sample of 31 local star-forming galaxies from the COS Legacy Archive Spectroscopic SurveY (CLASSY), selected as analogues of high-$z$ systems. Using co-spatial UV absorption and optical emission-line spectroscopy, we compare abundances of N, O, S, and Fe in the low-ionization (neutral) and high-ionization (ionized) gas, providing a multi-phase view of enrichment shortly after the current starburst and over longer timescales when ejecta from previous episodes have cooled and mixed. We find that O and S, produced predominantly in short-lived massive stars, are well mixed between the two phases, with scatter reflecting local inhomogeneities. Fe, predominantly produced by Type Ia supernovae on $\sim$1 Gyr timescales, is higher in the neutral gas, reflecting either delayed mixing of older Fe-enriched material or preferential depletion of Fe from the ionized phase through dust formation in core-collapse supernova ejecta. N exhibits the largest phase offset, with N/H$_{ion}$ systematically $\sim$0.7 dex higher than N/H$_{neu}$, and the magnitude of this offset correlates with stellar mass, metallicity, star-formation rate, and most strongly with the ISM outflow velocity. N/O ratios in the ionized phase rise rapidly within 3-6 Myr relative to the neutral gas, consistent with N enrichment dominated by Wolf-Rayet stars rather than intermediate-mass AGB stars on longer timescales. These results demonstrate that localized stellar feedback, outflows, and phase-dependent mixing collectively regulate the chemical evolution of star-forming galaxies, providing key insight into the extreme N/O abundances recently observed in galaxies at cosmic dawn.