The effect of curvature on the band alignment of one-dimensional (1D) van der Waals (vdW) transition metal dichalcogenide (TMDC) heterostructures is studied by comprehensive first-principles calculations. We find that, as the diameter of a TMDC nanotube decreases, the combined effect of curvature-induced flexoelectricity and circumferential tensile strain causes a rapid lowering of the conduction band minimum, whereas the valence band maximum exhibits an initial lowering before rising. As individual TMDC nanotubes form coaxial heterostructures, the concerted effect of diameter-dependent band-edge levels and intertube coupling via flexovoltage can result in a transition of intertube band alignment from Type II to Type I in multiple heterostructural systems, including large-diameter MoSe2@WS2, MoTe2@MoSe2, and MoTe2@WS2 heterostructures. These results lay down a foundation for the rational design of 1D vdW heterostructures.