We present the results of a multiwavelength study of the 19 most significant submillimetre (submm) sources detected in the SCUBA 8-mJy survey. As described in Scott et al., this survey covers similar or equal to 260 arcmin(2) using the submillimetre camera SCUBA, to a limiting source detection limit S-850 (mum) similar or equal to 8 mJy. One advantage of this relatively bright flux-density limit is that accurate astrometric positions are potentially achievable for every source using existing radio and/or millimetre-wave interferometers. However, an associated advantage is that spectral energy distribution (SED) based redshift constraints should be more powerful than in fainter submm surveys. Here we therefore exploit the parallel SCUBA 450-mum data, in combination with existing radio and Infrared Space Observatory (ISO) data at longer and shorter wavelengths to set constraints on the redshift of each source. We also analyse new and existing optical and near-infrared imaging of our SCUBA survey fields to select potential identifications consistent with these constraints. Our derived SED-based redshift constraints, and the lack of statistically significant associations with even moderately bright galaxies allow us to conclude that all 19 sources lie at z > 1, and at least half of them apparently lie at z > 2.