If you're talking about a multi-turn coil, that's not a simple calculation. There are a lot of variables. First consideration is what the current will be. Then you can determine the heat loss, then you have to figure out how quickly the heat will dissipate - this will depend on the coil geometry, conductor size, insulation, etc.
There may be some empirical methods for estimating this - motor and transformer designers must deal with similar problems fairly often.
The initial temperature rise rate is simple. The input power to the coil is I^2R in Watts=Joules/second. That power is then heating the copper which has a mass and specific heat.
The current will decrease as the coil heats up due to the increased resistance of the copper with temperature.
The final temperature, as dpc says, is tough to determine. I often use the natural convection heat loss in free air as a rough "guesstamate."
The time to temperature is then approximately five times the intercept time between the temperature rise line and the final temperature line.
