The newly developed Nematic Bond Theory enables an efficient way of studying phases and phase transitions in classical frustrated magnets with Heisenberg interactions. We extend here the Nematic Bond Theory to hold for non-Bravais lattices, and develop in addition an approximate expression for the free energy, which can be calculated at low computational cost. Using these extensions, we study classical frustrated Heisenberg models on triangular and honeycomb lattices. On the triangular lattice, we derive the phase diagram for the J1-J2-J3 Heisenberg model with ferromagnetic nearest neighbour interactions. In addition to detect lattice-nematic phases, we also find a novel symmetry broken state, which shows an extended maximum in the spin correlation function. We conjecture that this state is caused by domain wall excitations. The honeycomb lattice is studied to benchmark the Nematic Bond Theory on a non-Bravais lattice. For the antiferromagnetic J1-J2 Heisenberg model, we find that the Nematic Bond Theory shows good agreement with the literature for both symmetry broken lattice-nematic phases and symmetric spin liquid states.