We present 1.3 and 3 mm aperture synthesis imaging of the multiple T Tauri system UZ Tauri. UZ Tau is a hierarchical triple composed of a single star, UZ Tau E, 530 AU distant from a 50 AU binary, UZ Tau W. Both dust and gas emission from the close binary are at least a factor of four lower than from the single star. Since UZ Tau E and W have similar stellar masses, luminosities, and ages, we conclude that the mass of dust and gas associated with UZ Tau W is reduced solely by the influence of a close companion. The disk emission from UZ Tau E is best interpreted as a circumstellar disk similar to those around other single T Tauri stars. In a 1"-resolution aperture synthesis map, CO (2-->1) emission is coincident with the continuum peak and elongated with a size of 300 AU (FWHM); a velocity gradient is seen along the long axis, consistent with rotation in a gaseous disk. The emission is elongated at position angle 19 degrees, the same as the PA of previous polarization measurements. A disk model fit to the continuum spectral energy distribution (SED) of UZ Tau E yields a disk mass of 0.06 M_sun. In contrast, no CO emission is detected from UZ Tau W, and its 1.3 mm continuum emission is unresolved in a 1" (FWHM) beam (corresponding to a 70 AU radius). The small extent of the emission and dynamical considerations imply that the 50 AU binary cannot be surrounded by any appreciable circumbinary disk; its mm-wave emission is from circumstellar disks around one or both components. The mass of the circumstellar material is in the range 0.002--0.04 M_sun; the large uncertainty is due to the unknown temperature and surface density distributions of the material. The properties of the UZ Tau E disk are similar to those inferred for the early solar nebula; such a disk could give rise to a planetary system like our own. The mass of the UZ Tau W disk(s) is only marginally consistent with a ``minimum mass solar nebula.'' The constraints on disk size in UZ Tau W indicate that reduced mm-wave flux may be linked to a disparity in the size of disks (and therefore of planetary systems) around single and binary stars.