What is the time-constant relation between an electromagnet and its coil?
An inductor has a time constant, L/R, where R is the resistance of the windings. This is how long it will take for the current to reduce to 27% of its initial value, when the coil is short-circuited at the terminals.

For the same coil of wire, the inductance in air will be less than the inductance when assembled onto the iron core of an electromagnet. The inductance would increase further if the electromagnet had some iron on, or in the vicinity of, the pole pieces, to further reduce the air-gap in the magnetic circuit. Obviously, as the inductance changes, the time constant changes.