We present a fully relativistic investigation of the radiative recombination of a twisted electron with a bare heavy nucleus. The twisted electron is described by the wave function which accounts for the interaction with the nucleus in all orders in αZ. We use this wave function to derive the probability of the radiative recombination with a single ion being shifted from the twisted electron propagation direction. This probability is utilized for the consideration of a more realistic experimental scenario where the target is infinitely wide (macroscopic). The situation when the incident electron is a coherent superposition of two vortex states is considered as well. For the nonrelativistic case we present analytical expressions which support our numerical calculations. We study in details the influence of the electron twistedness on the polarization and angular distribution of the emitted photon.