We present parallaxes, proper motions, and J-band photometry for 348 L and T dwarfs measured using the wide-field near-infrared camera (WFCAM) on the United Kingdom Infrared Telescope (UKIRT). This is the largest single batch of infrared parallaxes for brown dwarfs to date. Our parallaxes have a median uncertainty of 3.5mas, similar to most previous ground-based infrared parallax surveys. Our target list was designed to complete a volume-limited parallax sample of L0-T8 dwarfs out to 25pc spanning declinations -30{deg} to +60{deg} (68% of the sky). We report the first parallaxes for 165 objects, and we improve on previous measurements for another 53 objects. Our targets include 104 objects (mostly early-L dwarfs) having Gaia DR2 parallax measurements with which our parallaxes are consistent. We include an extensive comparison of previous literature parallaxes for L and T dwarfs with both our results and Gaia DR2 measurements, identifying systematic offsets for some previous surveys. Our parallaxes confirm that 14 objects previously identified as wide common proper motion companions to main-sequence stars have distances consistent with companionship. We also report new J_MKO_ photometry for our targets, including the first measurements for 193 of our targets and improvements over previously published J_MKO_ photometry for another 60 targets. Altogether, our parallaxes will enable the first population studies using a volume-limited sample spanning spectral types L0-T8 defined entirely by parallaxes.