We report the first star formation history study of the Milky Ways nuclear star cluster (NSC), which includes observational constraints from a large sample of stellar metallicity measurements. These metallicity measurements were obtained from recent surveys from Gemini and the Very Large Telescope of 770 late-type stars within the central 1.5pc. These metallicity measurements, along with photometry and spectroscopically derived temperatures, are forward modeled with a Bayesian inference approach. Including metallicity measurements improves the overall fit quality, as the low-temperature red giants that were previously difficult to constrain are now accounted for, and the best fit favors a two-component model. The dominant component contains 93%+/-3% of the mass, is metal-rich (\overline{[M/H]}~0.45), and has an age of 5_-2-^+3^Gyr, which is ~3Gyr younger than earlier studies with fixed (solar) metallicity; this younger age challenges coevolutionary models in which the NSC and supermassive black holes formed simultaneously at early times. The minor population component has low metallicity (\overline{[M/H]}~-1.1) and contains ~7% of the stellar mass. The age of the minor component is uncertain (0.1-5Gyr old). Using the estimated parameters, we infer the following NSC stellar remnant population (with ~18% uncertainty): 1.5x10^5^ neutron stars, 2.5x10^5^ stellar-mass black holes (BHs), and 2.2x10^4^ BH-BH binaries. These predictions result in 2-4 times fewer neutron stars compared to earlier predictions that assume solar metallicity, introducing a possible new path to understand the so-called "missing-pulsar problem". Finally, we present updated predictions for the BH- BH merger rates (0.01-3Gpc^-3^/yr).