When charge moves at a constant speed, it is said that it produces static magnetic and electric field.
Why static ? If it moves, at every points, electric field and magnetic field change as they are dependent on a distance to the point. So even if charge moves at constant speed, we still should have electric and magnetic fields changing in every point in space.
Indeed, neither the electric nor the magnetic fields are static in the case of a charge moving with constant velocity. However, the fields, although they are not static, they also do not radiate.
Change does not mean radiation, to get radiation a necessary condition is that the field components vary as $\mathcal O(r^{-1})$, Then $S=E\times B$ will be $\mathcal O(r^{-2})$ which will give a nonzero value when multiplied by $4\pi r^2$, a surface area far away. This cannot happen if there is no work done on the source, see, energy conservation. A charge moving with a constant velocity all by itself is neither doing work nor having work done on.
