The linear polarization of the characteristic photon emission from few-electron ions is studied for its sensitivity with regard to the nuclear spin and magnetic moment of the ions. Special attention is paid, in particular, to the K$alpha_1$ ($1s 2p_{
3/2} ^{1,3}P_{1,2} to 1s^2 ^1S_0$) decay of selected helium-like ions following the radiative electron capture into initially hydrogen-like species. Based on the density matrix theory, a unified description is developed that includes both, the many-electron and hyperfine interactions as well as the multipole-mixing effects arising from the expansion of the radiation field. It is shown that the polarization of the K$alpha_1$ line can be significantly affected by the mutipole mixing between the leading $M2$ and hyperfine-induced $E1$ components of $1s2p ^3P_2, F_i to 1s^2 ^1S_0, F_f$ transitions. This $E1$-$M2$ mixing strongly depends on the nuclear properties of the considered isotopes and can be addressed experimentally at existing heavy-ion storage rings.
