During the last decades, the electron impact excitation (EIE) process has aroused much interest in various research areas. This process is crucial to the diagnoses of astrophysical and laboratory plasmas. Moreover, the EIE studies play an important role in understanding the quantum electrodynamic, many-electron, and hyperfine interaction effects in heavy atomic systems. As is well known, when ions are excited by collisions with a unidirectional beam of electrons, the magnetic sublevels of the excited state may be populated with nonstatistical probability. In the decay of the excited state, the emitted radiation is found to be anisotropic and polarized. From the analysis of the polarization, valuable information can be obtained. These properties have become indispensable tools for the diagnosis of plasma state and the analysis of complex spectrum formation mechanism. Up to now, however, most of studies have dealt with the linear polarization of X-ray radiation. Fewer publications have reported the circular polarization. Moreover, theoretical studies of the characteristic X-ray emission have just dealt with ions having zero nuclear spin, or have simply omitted all contributions that arise from such a spin. It is known that some kinds of ions each have a nuclear spin I 0. Owing to the hyperfine coupling, new decay channel will be open, namely, hyperfine-induced transition. It is thus important to analyze how the hyperfine interaction affects the polarization properties of X-ray radiation. In this study, we present a systematically theoretical analysis of the polarization and angular distribution of X-ray radiation during the hyperfine-induced transition. The calculations are performed by using a fully relativistic distorted-wave method. Special attention is paid to the studies of angular correlations and polarization properties of the 1s2p 3P2 Fi=3/2 1s2 1S Ff=1/2 decay for highly charged He-like Sc19+ and 205Tl79+ ions with nuclear spin I=1/2 following impact excitation by a completely longitudinally-polarized electron beam. Two effects, i.e.the BI and the mutipole mixing between the leading M2 decay and hyperfine-induced E1 decay, on the polarization of the emitted radiation are discussed. Our results show that both the BI and the E1-M2 interference effects may significantly affect the polarization and angular emission pattern of the transition line. For example, the BI and the E1-M2 mixing lead the circular polarization to increase by about 50% and 40% for 205Tl79+ ions, respectively. With the development of the X-ray detectors, the measurement on the polarization during the hyperfine-induced transition becomes feasible. We hope that the present results would be useful in resolving some disagreement between the theories and experiments relating to the polarization properties of the X-ray radiation.