Spintronic terahertz (THz) emitters based on ferromagnet/heavy-metal heterostructures provide efficient broadband sources of THz radiation, yet the microscopic relation between ultrafast spin transport and charge redistribution remains incompletely understood. Here, we employ time-dependent density functional theory to resolve the coupled charge, spin, and magnetization dynamics of Co/Pt and Co/W bilayers in real time and space. Despite qualitatively similar interlayer spin-current dynamics, the two heterostructures exhibit markedly different interlayer charge transfer: Co/Pt shows a relatively small and partially reversed transfer from Pt to Co, whereas Co/W exhibits a larger and persistent transfer from Co to W. We trace these differences to the material-dependent ground-state electronic structure and the spin- and orbital-resolved phase space accessible to photoexcited carriers. Spin-resolved dynamics further reveal coexisting spin-orbit-mediated redistribution and optically induced intersite spin transfer, including a delayed minority-spin transfer from the heavy metal to Co common to both systems. While net electronic spin angular momentum is transferred between the layers, we find no corresponding net transfer of electronic orbital angular momentum. In Co/W, the persistent redistribution of electronic density from interstitial to atom-centered states additionally provides strong evidence for light-induced electronic localization. These results demonstrate that ultrafast spin transport does not uniquely determine charge or angular-momentum redistribution in spintronic THz emitters, which instead depends critically on the material-specific electronic phase space.