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 μm 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 indicate T_eff = 1300 K and [M/H] ∼ ‑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 μm 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.