It is the changing magnetic and electric fields. The radiated fields can be calculated from the changing electric (or magnetic) currents. The fields at some distance away are calculated from potentials that are the sum of contributions from all the currents, delayed by the time it takes light to get there from each little bit of current. The fields are obtained from the retarded potentials.
If the current doesn't change, neither do the magnetic and electric fields, and there is no radiation, just a static electric and/or magnetic field.
Radiation from a wire is small compared to the power for the current that is usually in a wire for circuits with impedances above a few ohms. Wire antennas are usually made resonant to increase the current on the wire so that the radiation is much larger.