How would magnetic wire (not magnet wire) affect the inductance of a coil?
If the structure is identical except for the wire you would get a slightly higher inductance reading as the "core" would now include not only your inductor core but the permeability of the wire.

The Mu Metal Wire you mention is not used to wind the core. It's usually used for magnetic "shielding". You need a material that offers a low reluctance path for magnetic fields to shape them (since you can't shield a magnetic field, only redirect it).