Recently, new observations of 6 Li in Population II stars of the Galactic halo have shown a surprisingly high abundance of this isotope, about a thousand times higher than its predicted primordial value. In previous papers, a cosmological model for the cosmic-ray-induced production of this isotope in the intergalactic medium (IGM ) has been developed to explain the observed abundance at low metallicity. In this paper, given this constraint on the 6 Li, we calculate the nonthermal evolution with redshift of D, Be, and B in the IGM. In addition to cosmological cosmic ray interactions in the IGM, we include additional processes driven by supernova explosions: neutrino spallation and a low-energy component in the structures ejected by outflows to the IGM. We take into account CNO CRs impinging on the intergalactic gas. Although subdominant in the Galactic disk, this process is shown to produce the bulk of Be and B in the IGM, due to the differential metal enrichment between structures (where CRs originate) and the IGM. We also consider the resulting extragalactic gamma-ray background, which we find to be well below existing data. The computation is performed in the framework of hierarchical structure formation, considering several star formation histories, including Population III stars. We find that D production is negligible and that a potentially detectable Be and B plateau is produced by these processes at the time of the formation of the Galaxy (z $ 3).