It is demonstrated that shaped topological insulator (TI) nanowires, i.e., such that their cross-section radius varies along the wire length, can be tuned into a number of different transport regimes when immersed in a homogeneous coaxial magnetic field. This is in contrast with widely studied tubular nanowires with constant cross section, and is due to magnetic confinement of Dirac surface carriers. In flat two-dimensional systems, such a confinement requires inhomogeneous magnetic fields, while for shaped nanowires of standard size homogeneous fields of the order of B∼1T are sufficient. We put recent work [R. Kozlovsky et al., Phys. Rev. Lett. 124, 126804 (2020)] into broader context and extend it to deal with axially symmetric wire geometries with arbitrary radial profile. A dumbbell-shaped TI nanowire is used as a paradigmatic example for transport through a constriction and shown to be tunable into five different transport regimes: (i) conductance steps, (ii) resonant transmission, (iii) current suppression, (iv) Coulomb blockade, and (v) transport through a triple quantum dot. Switching between regimes is achieved by modulating the strength of a coaxial magnetic field and does not require strict axial symmetry of the wire cross section. As such, it should be observable in TI nanowires fabricated with available experimental techniques.