The Joint Astronomy Centre operates two telescopes at the Mauna Kea Observatory: the James Clerk Maxwell Telescope, operating in the submillimetre, and the United Kingdom Infrared Telescope, operating in the near and thermal infrared. Both wavelength regimes benefit from the ability to schedule observations flexibly according to observing conditions, albeit via somewhat different "site quality" criteria. Both UKIRT and JCMT now operate completely flexible schedules. These operations are based on telescope hardware which can quickly switch between observing modes, and on a comprehensive suite of software (ORAC/OMP) which handles observing preparation by remote PIs, observation submission into the summit database, conditions-based programme selection at the summit, pipeline data reduction for all observing modes, and instant data quality feedback to the PI who may or may not be remote from the telescope. This paper describes the flexible scheduling model and presents science statistics for the first complete year of UKIRT and JCMT observing under the combined system.