Elemental composition guides the formation, evolution, and atmosphere processes of the lowest-mass stars and brown dwarfs, or ultracool dwarfs (UCDs). While the vast majority of known UCDs are solar-metallicity objects in the Milky Way's thin disk, a rare subsample of metal-poor stellar and substellar subdwarfs are known in the Solar Neighborhood. We report the first results from our 0.6-14 µm spectroscopic survey of 32 L and T dwarfs and subdwarfs, including 11 wide companions to FGK stars, aimed at investigating the influence of metallicity on chemistry, condensation, and disequilibrium mixing in the coolest atmospheres. We show the first temperature and metallicity spectral grid for ultracool dwarfs spanning 0.6-14 µm, and identify key metallicity-sensitive features in this regime including evidence of PH3 absorption in the moderate-resolution spectrum of the metal-poor companion T dwarf Wolf 1130C. We demonstrate robust fits to these data to select model grids, including the first self-consistent fits to the extreme T subdwarfs J1810-1010 and J0414-5854 with the SAND model grid, and report preliminary atmosphere parameters from NIRSpec+MIRI low-resolution spectra and radial velocities from moderate-resolution NIRSpec spectra. This program is supported by NASA through program JWST-GO-04688.