The current being supplied by the battery is limited by the internal resistance of the battery itself. If this is low, the current flowing in a short circuit or near short circuit will be on the high side.
The wire will, in essence, act as a resistor, albeit a low value one, and not as a true short, as will that of the internal resistance of the battery. It is this that will define the actual speed of current increase.
Obviously, the lower the resistance of the circuit, the higher the current will be allowed to flow through that circuit and the lower the time taken will be to get it there.
Also remember that the current will be the same in all parts of the circuit, so it is not only the wire being subjected to the high current, but in the battery also.
rgds