We investigate the nature of star formation in gas-rich galaxies at z > 7 forming in a markedly overdense region, in the vicinity of a massive virialized halo already exceeding 1012 M⊙, through the use of the very-high-resolution cosmological hydrodynamical simulation MassiveBlackPS with a spatial resolution of 2 pc. We find that not only the primary galaxy but also the lower-mass companion galaxies rapidly develop massive self-gravitating compact gas disks, less than 500 pc in size, which undergo fragmentation by gravitational instability into very massive bound clumps. Star formation proceeds fast in the clumps, which quickly turn into compact star clusters with masses in the range 105-108 M⊙ and typical half-mass radii of a few parsec, reaching characteristic densities above 105 M⊙ pc−2. The properties of the clusters in the lowest-mass galaxy bear a striking resemblance to those recently discovered by the James Webb Space Telescope (JWST) in the lensed Cosmic Gems arc system at z = 10.2. We argue that, due to their extremely high stellar densities, intermediate-mass black holes would form rapidly inside the clusters, which would then swiftly sink and merge on their way to the galactic nucleus, easily growing into a 107 M⊙ supermassive black hole (SMBH). Due to the high fractional mass contribution of clusters to the stellar mass of the galaxies, in the range 20%-40%, the central SMBH would comprise more than 10% of the mass of its host galaxy, naturally explaining the overmassive SMBHs discovered by JWST at z > 6.