  We report JWST/NIRSpec spectra of three distant T-type brown dwarfs
identified in the Ultradeep NIRSpec and NIRCam ObserVations before the Epoch of
Reionization (UNCOVER) survey of the Abell 2744 lensing field. One source was
previously reported as a candidate T dwarf on the basis of NIRCam photometry,
while two sources were initially identified as candidate active galactic
nuclei. Low-resolution 1--5 $\mu$m spectra confirm the presence of molecular
features consistent with T dwarf atmospheres, and comparison to spectral
standards infers classifications of sdT1, T6, and T8--T9. The warmest source,
UNCOVER-BD-1, shows evidence of subsolar metallicity, and atmosphere model fits
indicates T$_{eff}$ = 1300 K and [M/H] $\sim$ $-$1.0, making this one of the
few spectroscopically-confirmed T subdwarfs known. The coldest source,
UNCOVER-BD-3, is near the T/Y dwarf boundary with T$_{eff}$ = 550 K, and our
analysis indicates the presence of PH$_3$ in the 3--5~$\mu$m region, favored
over CO$_2$ and a possible indicator of subsolar metallicity. We estimate
distances of 0.9--4.5 kpc from the Galactic midplane, making these the most
distant brown dwarfs with spectroscopic confirmation. Population simulations
indicate high probabilities of membership in the Galactic thick disk for two of
these brown dwarfs, and potential halo membership for UNCOVER-BD-1. Our
simulations indicate that there are approximately 5 T dwarfs and 1--2 L dwarfs
in the Abell 2744 field down to F444W = 30 AB mag, roughly one-third of which
are thick disk members. These results highlight the utility of deep
JWST/NIRSpec spectroscopy for identifying and characterizing the oldest
metal-poor brown dwarfs in the Milky Way.
