The proximity-induced spin-orbit coupling (SOC) in heterostructures of twisted graphene and topological insulators (TIs) Bi2Se3 and Bi2Te3 is investigated from first principles. To build commensurate supercells, we strain graphene and correct thus resulting band offsets by applying a transverse electric field. We then fit the low energy electronic spectrum to an effective Hamiltonian that comprises orbital and spin-orbit terms. For twist angles 0 degrees 0 <= 20 degrees, we find the dominant spin-orbit couplings to be of the valley-Zeeman and Rashba types, both a few meV strong. We also observe a sign change in the induced valley-Zeeman SOC at 0 approximate to 10 degrees. Additionally, the in-plane spin structure resulting from the Rashba SOC acquires a nonzero radial component, except at 0 degrees or 30 degrees. At 30 degrees the graphene Dirac cone interacts directly with the TI surface state. We therefore explore this twist angle in more detail, studying the effects of gating, TI thicknesses, and lateral shifts on the SOC parameters. We find, in agreement with previous results, the emergence of the proximitized Kane-Mele SOC, with a change in sign possible by electrically tuning the Dirac cone within the TI bulk band gap.