We present submillimetre (sub-mm) photometry for 11 hyperluminous infrared galaxies (HLIRGs, ) and use radiative transfer models for starbursts and active galactic nuclei (AGN) to examine the nature of the IR emission. In all the sources both a starburst and AGN are required to explain the total IR emission. The mean starburst fraction is 35 per cent, with a range spanning 80 per cent starburst-dominated to 80 per cent AGN-dominated. In all cases the starburst dominates at rest-frame wavelengths longwards of 50 mum, with star formation rates >500 Msolar yr-1. The trend of increasing AGN fraction with increasing IR luminosity observed in IRAS galaxies is observed to peak in the HLIRG population, and not increase beyond the fraction seen in the brightest ultraluminous infrared galaxies (ULIRGs). The AGN and starburst luminosities correlate, suggesting that a common physical factor, plausibly the dust masses, govern the luminosities of starbursts and AGN in HLIRGs. Our results suggest that the HLIRG population is composed both of ULIRG-like galaxy mergers and of young galaxies going through their maximal star formation periods whilst harbouring an AGN. The detection of coeval AGN and starburst activity in our sources implies that starburst and AGN activity, and the peak starburst and AGN luminosities, can be coeval in active galaxies generally. When extrapolated to high z our sources have comparable observed frame sub-mm fluxes to sub-mm survey sources. At least some high-z sub-mm survey sources are therefore likely to be composed of similar galaxy populations to those found in the HLIRG population. It is also plausible from these results that high-z sub-mm sources harbour heavily obscured AGN. The differences in X-ray and sub-mm properties between HLIRGs at z~ 1 and sub-mm sources at ~3 implies some level of evolution between the two epochs. Either the mean AGN obscuration level is greater at z~ 3 than at z~ 1, or the fraction of IR-luminous sources at z~ 3 that contain AGN is smaller than that at z~ 1.