From what approximate size does the collapsing iron core shrink to form a compact object?

From a size comparable to Earth down to just a few tens of kilometers.

The gravitational collapse following the loss of core pressure drives a dramatic reduction in the physical scale of the stellar remnant. Before the density reaches the point where nuclear matter resists further compression—a density similar to that found within an atomic nucleus—the core undergoes massive shrinkage. Specifically, the core compresses from a size comparable to that of the Earth down significantly to a diameter measured in just tens of kilometers. This incredible compaction leads to astronomical increases in density, setting the stage for either the formation of a neutron star or, if the mass is too great, a black hole.