Well, this popped inside my head when I was doing Boit-Savart's law.
$$d\vec{B}=\frac{\mu_{0}I d\vec{l} \times \vec{r}}{r^3}$$
l is the vector that represents the current element (i.e the direction of current flow) and r represents the point at which we have to find the magnetic field. So from this can we infer that the Magnetic field inside a conductor is uniform as the $\vec{l}$ and $\vec{r}$ are in the same direction and thus the cross product is 0. So dB=0 and thus B is uniform along the vector that we have taken…
Elaborated: Take a point P inside the conductor. Now take another point Q such that $\vec{PQ}$ is parallel to the surface. Let $P$ and $Q$ be at a distance $a$ from the center. Then $\vec{r}$ and $\vec{l}$ are in the same line $\vec{PQ}$ making the angle between them zero. So $dB=0$. Therefore $B$ remains uniform along the vector $\vec{PQ}$.
Best Answer
For DC current the field increases from zero on the axis of the conductor to a maximum value on the surface. It is not uniform across the cross section of the conductor. Of course, it is constant in the sense that there is no time dependence (what "constant" usually means).