We have investigated the formation of globular cluster (GC) systems in the fiducial semi-analytic model of galaxy formation of Cole et al., by assuming that GCs are formed at high redshifts (z>5) in protogalactic fragments, and during the subsequent gas-rich merging of these fragments. Under these assumptions we have simulated the GC systems of 450 elliptical galaxies, and find that the majority (93 per cent) are intrinsically bimodal in metallicity. We find that, in the mean, the metal-rich GC subpopulations are younger than the metal-poor GC subpopulations, with ages of 9 and 12Gyr respectively, and that the mean ages of the metal-rich GCs are dependent upon host galaxy luminosity and environment (halo circular velocity), whereas the metal-poor GCs are not. We find that the continued gaseous merging of the protogalactic fragments leads to significant age-structure amongst the metal-rich GCs. These GCs exhibit a large age-range (5 to 12Gyr), which increases for low-luminosity galaxies, and for galaxies in low circular velocity haloes. Moreover, the metal-rich GCs associated with low-luminosity field and/or group ellipticals are ~2Gyr younger than the metal-rich GCs in luminous cluster ellipticals. We find that the total GC populations scale with host galaxy luminosity as NGC\proptoL1.25V,gal, a result in agreement with observations of luminous elliptical galaxies. This scaling is due to a systematic increase in the M/L ratios of the galaxy haloes with luminosity for LV,gal>L* galaxies in the model. A comparison between the luminosity growth of the model ellipticals and their GC formation indicates that mergers do not significantly affect SN at z<2. We find the mean colours of the both the metal-rich and metal-poor GCs exhibit only a weak dependence upon host galaxy luminosity, a result consistent with contemporary observations. We conclude that gaseous merging, the bulk of which occurs at 1<=z<=4 in our ΛCDM model, leads to the formation of the metal-rich peak of the GC systems of elliptical galaxies. We suggest that the formation and subsequent truncation of the metal-poor GCs in the protogalactic fragments is closely related to the star formation rate in these fragments, which may have been significantly higher at very early times.