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We present a pioneering, spatially resolved, multiphase gas abundance study on the blue compact dwarf galaxy NGC 5253, targeting 10 star-forming (SF) clusters inside six far-UV (FUV) Hubble Space Telescope/Cosmic Origins Spectrograph (COS) pointings with cospatial optical Very Large Telescope/MUSE observations throughout the galaxy. The SF regions span a wide range of ages (1–15 Myr) and are distributed at different radii (50–230 pc). We performed a robust absorption-line profile fitting on the COS spectra, covering 1065–1430 Å in the FUV, allowing an accurate computation of neutral-gas abundances for 13 different ions sampling eight elements. These values were then compared with the ionized-gas abundances, measured using the direct method on MUSE integrated spectra inside analog COS apertures. Our multiphase, spatially resolved comparisons find abundances, which are lower in the neutral gas than the ionized gas by 0.22, 0.80, and 0.58 dex for log(O/H), log(N/H), and log(N/O), respectively. We modeled the chemical abundance distributions and evaluated correlations as a function of the radius and age. It was found that, while N, O, and N/O abundances decrease as a function of age in the ionized gas, they increase with age in the neutral gas. No strong correlations for N, O, or N/O were observed as a function of the radius. The N/O and N/H offsets between the phases were found to decrease with age, providing evidence that chemical enrichment happens differentially, first in the ionized-gas phase around 2–5 Myrs (due to N-rich Wolf–Rayet stars) and then mixing out into the cold neutral gas on longer timescales of 10–15 Myr.