Motor rewinding - saturation concerns
Will there be saturation from iron or permanent magnets?

No.
Magnetic flux is directly proportional to current multiplied by turns. Assuming all the winding space is used, if you increase the number of turns then you have to decrease the wire diameter, which increases resistance and reduces current draw. The result will be less flux and torque than the original motor could produce on the same voltage.
For example if you double the number of turns then each turn has to fit in half the area, so its resistance will be double that of the original. Since you have doubled the number of turns, the total winding resistance will increase by 4 times and the current will be reduced to 1/4. The result is half the flux (and torque) of the original motor.
But what if you raise the voltage? Double the voltage and current is doubled to half that of the original winding, and flux and torque are restored. Kv was halved by having twice as many turns, but the higher voltage makes up for it. Power dissipated in the windings is also the same, so you get back to the same performance as the original motor but at double the voltage and half the current.
To get more torque you have to go the other way and reduce the number of turns, which increases current. Go too far and the flux produced may be enough to saturate the core, then any further current increase has less effect. Theoretically the flux could get high enough to demagnetize the magnets, but the current required to demagnetize Neodym magnets would probably melt the windings.
The real danger is heat from the windings raising magnet temperature enough to  demagnetize them. This is not just theoretical, as high power magnets may suffer partial demagnetization at temperatures as low as 80 °C.