We use a simple model of spheroid formation to explore the relationship between the creation of stars and dust in a massive protogalaxy and the growth of its central black hole. This model predicts that submillimetre luminosity peaks after only ~=0.2 Gyr. However, without a very massive seed black hole, Eddington-limited growth means that a black hole mass of 109 Msolar, and hence very luminous active galactic nuclei (AGN) activity, cannot be produced until >0.5 Gyr after the formation of the first massive stars in the halo. The model thus predicts a time-lag between the peak of submillimetre luminosity and AGN luminosity in a massive protoelliptical of a few times 108 yr. For a formation redshift z~= 5, this means that powerful AGN activity is delayed until z~= 3.5, by which time star formation in the host is ~=90 per cent complete, and submillimetre luminosity has declined to ~=25 per cent of its peak value. This provides a natural explanation for why successful submillimetre detections of luminous radio galaxies are largely confined to z > 2.5. Conversely the model also predicts that while all high-redshift luminous submillimetre-selected sources should contain an active (and growing) black hole, the typical luminosity of the AGN in such objects is ~=1000 times smaller than that of the most powerful AGN. This is consistent with the almost complete failure to detect submillimetre selected galaxies with existing X-ray surveys. Finally, the model yields a black hole-spheroid mass ratio, which evolves rapidly in the first Gyr, but asymptotes to ~=0.001-0.003 in agreement with results at low redshift. This ratio arises not because the AGN terminates star formation, but because fuelling of the massive black hole is linked to the total mass of gas available for star formation in the host.