We present the discovery of 15 new T2.5-T7.5 dwarfs (with estimated distances ~24-93pc), identified in the first three main data releases of the United Kingdom Infrared Telescope (UKIRT) Infrared Deep Sky Survey. This brings the total number of T dwarfs discovered in the Large Area Survey (LAS) (to date) to 28. These discoveries are confirmed by near-infrared spectroscopy, from which we derive spectral types on the unified scheme of Burgasser et al. Seven of the new T dwarfs have spectral types of T2.5-T4.5, five have spectral types of T5-T5.5, one is a T6.5p and two are T7-7.5. We assess spectral morphology and colours to identify T dwarfs in our sample that may have non-typical physical properties (by comparison to solar neighbourhood populations), and find that three of these new T dwarfs may have unusual metallicity, two may have low surface gravity, and one may have high surface gravity. The colours of the full sample of LAS T dwarfs show a possible trend to bluer Y - J with decreasing effective temperature, and some interesting colour changes in J - H and z - J (deserving further investigation) beyond T8. The LAS T dwarf sample from the first and second main data releases show good evidence for a good level of completion to J = 19. By accounting for the main sources of incompleteness (selection, follow-up and spatial) as well as the effects of unresolved binarity, Malmquist and Eddington bias, we estimate that there are 17 +/- 4 >= T 4 dwarfs in the J <= 19 volume of the LAS second data release. This value is most consistent with theoretical predictions if the substellar mass function exponent α (dN/dm ~ m-α) lies between -1.0 and 0. This is consistent with the latest 2-Micron All Sky Survey (2MASS)/Sloan Digital Sky Survey (SDSS) constraint (which is based on lower number statistics) and is significantly lower than the α ~ 1.0 suggested by L dwarf field populations, which is possibly a result of the lower mass range probed by the T dwarf class.