Help Understanding How Long It Will Take To Heat My Wire
You can reach any given temperature in nano-seconds if you ram in enough power(+).

Calculate the mass of the wire, length*area*specific_gravity. Look up the specific heat capacity of the wire material, Kaye and Laby online is my usual resource, and calculate the heat capacity of the wire. Note that the capacity will vary with temperature, so you'll need to use a mean figure, or do a numerical integration, if you want reasonable accuracy. Mean figure is probably OK for the accuracy needed. Multiply by the number of degrees you want to heat through, that gives you the heat in Joules you need to deliver during the heating phase. Now divide by your target heating time, that gives you your minimum power in watts for heating. This is a minimum figure, the adiabatic figure, as heat will be lost from the wire by radiation, and by conduction though the ends, during heating. To a first approximation, for a short heating time, this amount will be small. Pick a voltage or current that will deliver that power (for any given heater, once one is chosen, the other is automatically fixed).(++)

How is the wire temperature going to be regulated? If by temperature feedback, say from the wire resistance, then your controller will handle the transition from heating to staying warm OK. If by dead reckoning, simply using a defined voltage or current through the wire so that power in matches power out at the target temperature, then you may have a problem. It's likely the heating power will exceed the keep-hot power, if you want to heat fast. This means you'll need either a timer (to do it by dead reckoning), or a resistance measurement (OK if using steel for instance, more difficult if the wire is a low tempco resistance alloy wire) to switch from heating power to keep-hot power.

As a general rule, if you want to use a single power setting for both heating and keep-hot, as wire mass (heating time) goes by radius^2, and wire area (radiation loss, convection loss, keep-hot power) goes by radius^1, you'll get fastest heating finest wire you can use.

(+) Look up Sandia's Z-pinch machine, that makes wire quite hot, quite quickly.

(++) At this point, you're only one step away from computing the \$I^2t\$ of the wire, often specified for fuses and power semiconductors as a measure of what sort of pulse will get them to a given temperature (a max for survival, min for destruction).