We present a new sample of purely near-infrared-selected KVega < 16.5 [KAB < 18.4] extremely red [(J − K)Vega > 2.5] quasar candidates at z ∼ 2 from ≃900 deg2 of data in the UKIDSS Large Area Survey (LAS). Five of these are spectroscopically confirmed to be heavily reddened type 1 active galactic nuclei (AGN) with broad emission lines bringing our total sample of reddened quasars from the UKIDSS-LAS to 12 at z = 1.4–2.7. At these redshifts, Hα (6563 Å) is in the K band. However, the mean Hα equivalent width of the reddened quasars is only 10 per cent larger than that of the optically selected population and cannot explain the extreme colours. Instead, dust extinction of AV ∼ 2–6 mag is required to reproduce the continuum colours of our sources. This is comparable to the dust extinctions seen in submillimetre galaxies at similar redshifts. We argue that the AGN are likely being observed in a relatively short-lived breakout phase when they are expelling gas and dust following a massive starburst, subsequently turning into UV-luminous quasars. Some of our quasars show direct evidence for strong outflows (v ∼ 800–1000 km s−1) affecting the Hα line consistent with this scenario. We predict that a larger fraction of reddened quasar hosts are likely to be submillimetre bright compared to the UV-luminous quasar population. We use our sample to place new constraints on the fraction of obscured type 1 AGN likely to be missed in optical surveys. Taken at face value our findings suggest that the obscured fraction depends on quasar luminosity. The space density of obscured quasars is approximately five times that inferred for UV-bright quasars from the Sloan Digital Sky Survey (SDSS) luminosity function at Mi < −30 but seems to drop at lower luminosities even accounting for various sources of incompleteness in our sample. We find that at Mi ∼ −28 for example, this fraction is unlikely to be larger than ∼20 per cent although these fractions are highly uncertain at present due to the small size of our sample. A deeper K-band survey for highly obscured quasars is clearly needed to test this hypothesis fully and is now becoming possible with new sensitive all-sky infrared surveys such as the VISTA Hemisphere Survey and the Wide Infrared Survey Explorer (WISE) All Sky Survey.