The cubic chiral helimagnets with the $P2_13$ space group represent a group of compounds in which the stable skyrmion-lattice state is experimentally observed. The key parameter that controls the energy landscape of such systems and determines the emergence of a topologically nontrivial magnetic structures is the Dzyaloshinskii-Moriya interaction (DMI). Chemical substitution is recognized as a convenient instrument to tune the DMI in real materials and has been successfully utilized in studies of a number of chiral magnets, such as MnSi, FeGe, MnGe, and others. In our study, we applied small-angle neutron scattering to investigate how chemical substitution influences the skyrmionic properties of an insulating helimagnet Cu$_2$OSeO$_3$ when Cu ions are replaced by either Zn or Ni. Our results demonstrate that the DMI is enhanced in the Ni-substituted compounds (Cu,Ni)$_2$OSeO$_3$, but weakened in (Cu,Zn)$_2$OSeO$_3$. The observed changes in the DMI strength are reflected in the magnitude of the spin-spiral propagation vector and the temperature stability of the skyrmion phase.