We develop an accurate nanoelectronic modeling approach for realistic three-dimensional topological insulator nanostructures and investigate their low-energy surface-state spectrum. Starting from the commonly considered four-band k·p bulk model Hamiltonian for the Bi₂Se₃ family of topological insulators, we derive new parameter sets for Bi₂Se₃, Bi₂Te₃ and Sb₂Te₃. We consider a fitting strategy applied to ab initio band structures around the Γ point that ensures a quantitatively accurate description of the low-energy bulk and surface states, while avoiding the appearance of unphysical low-energy states at higher momenta, something that is not guaranteed by the commonly considered perturbative approach. We analyze the effects that arise in the low-energy spectrum of topological surface states due to band anisotropy and electron-hole asymmetry, yielding Dirac surface states that naturally localize on different side facets. In the thin-film limit, when surface states hybridize through the bulk, we resort to a thin-film model and derive thickness-dependent model parameters from ab initio calculations that show good agreement with experimentally resolved band structures, unlike the bulk model that neglects relevant many-body effects in this regime. Our versatile modeling approach offers a reliable starting point for accurate simulations of realistic topological material-based nanoelectronic devices. This dataset contains the data used in the corresponding publication.