The most massive galaxies in the present-day Universe are found to lie in the centres of rich clusters. They have old, coeval stellar populations suggesting that the bulk of their stars must have formed at early epochs in spectacular starbursts, which should be luminous phenomena when observed at submillimetre wavelengths. The most popular model of galaxy formation predicts that these galaxies form in proto-clusters at high-density peaks in the early Universe. Such peaks are indicated by massive high-redshift radio galaxies. Here we report deep submillimetre mapping of seven high-redshift radio galaxies and their environments. These data confirm not only the presence of spatially extended regions of massive star-formation activity in the radio galaxies themselves, but also in companion objects previously undetected at any wavelength. The prevalence, orientation, and inferred masses of these submillimetre companion galaxies suggest that we are witnessing the synchronous formation of the most luminous elliptical galaxies found today at the centres of rich clusters of galaxies.