Weakly Interacting Massive Particles (WIMPs) are a leading candidate for dark matter and are expected to produce nuclear recoil (NR) events within liquid xenon time-projection chambers. We present a measurement of the scintillation timing characteristics of liquid xenon in the LUX dark matter detector and develop a pulse shape discriminant to be used for particle identification. To accurately measure the timing characteristics, we develop a template-fitting method to reconstruct the detection times of photons. Analyzing calibration data collected during the 2013-16 LUX WIMP search, we provide a new measurement of the singlet-to-triplet scintillation ratio for electron recoils (ER) below 46~keV, and we make a first-ever measurement of the NR singlet-to-triplet ratio at recoil energies below 74~keV. We exploit the difference of the photon time spectra for NR and ER events by using a prompt fraction discrimination parameter, which is optimized using calibration data to have the least number of ER events that occur in a 50\% NR acceptance region. We then demonstrate how this discriminant can be used in conjunction with the charge-to-light discrimination to possibly improve the signal-to-noise ratio for nuclear recoils.