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How long does it take for the magnetic domains of a piece of iron to align (and form a magnetic field) from the moment it enters in contact with a magnetic field?

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The magnetization in the iron starts precessing immediately at an angular frequency given by $\gamma H$. The gyromagnetic ratio, $\gamma$, has a value of about 28 GHz per Tesla. Eventually the oscillation dies out due to various damping phenomena in the material and the magnetization tends to align with the field. In most cases this takes just a few cycles. Magnetic devices can typically operate at a few GHz - in hard disk drives, for example.

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