What speeds can the collapsing iron core reach as gravity takes over?

Nearly a quarter of the speed of light.

Once the iron core forms and the internal thermal pressure supporting the star vanishes instantaneously, gravity exerts its full, unchecked force, causing the core to plunge inward catastrophically. This inward collapse is extremely rapid, reaching immense velocities that are a significant fraction of the universal speed limit. The core shrinks dramatically, from an initial size similar to that of the Earth down to only a few tens of kilometers across. During this profound collapse phase, the infalling material achieves speeds approaching nearly one-quarter of the speed of light before the densities become high enough to halt the compression suddenly.