A conductor carries 10 Amps, and generates a magnetic field $B$.
How fast does it take for that current to flow from the wire throughout the entire circuit?
And how fast does it take for the magnetic field to be created?
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There are 2 ways of looking at the speed of an electric current.
The electrons themselves move quite slowly: of the order of 1.0 m/hour. In other words, if you have a 1m long cable, an electron will take a whole hour to get from one end to the other. For your example, in a copper wire of radius 1 mm carrying a steady current of 10 Amps, the drift velocity is only about 0.024 cm/sec.
However, as soon as an electron starts to move, it affects other electrons around it. Because of this, the current flows out of the other end of the cable almost immediately. As a result, electricity travels down the wire at about $2/3$ of the speed of light.
Magnetic fields, on the other hand, travel through space at the same speed as electric fields: the speed of light. I don't know how fast they travel through magnetic materials, but it will be somewhat slower than $c$.