  We report the first star formation history study of the Milky Way's nuclear
star cluster (NSC) that includes observational constraints from a large sample
of stellar metallicity measurements. These metallicity measurements were
obtained from recent surveys from Gemini and VLT of 770 late-type stars within
the central 1.5 pc. 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%$\pm$3% of the mass, is metal-rich
($\overline{[M/H]}\sim$0.45), and has an age of 5$^{+3}_{-2}$ Gyr, which is
$\sim$3 Gyr younger than earlier studies with fixed (solar) metallicity; this
younger age challenges co-evolutionary models in which the NSC and supermassive
black holes formed simultaneously at early times. The minor population
component has low metallicity ($\overline{[M/H]}\sim$ -1.1) and contains
$\sim$7% of the stellar mass. The age of the minor component is uncertain (0.1
- 5 Gyr old). Using the estimated parameters, we infer the following NSC
stellar remnant population (with $\sim$18% uncertainty): 1.5$\times$10$^5$
neutron stars, 2.5$\times$10$^5$ stellar mass black holes (BHs) and
2.2$\times$10$^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
(0.01-3 Gpc$^{-3}$yr$^{-1}$).
