The forward current though any diode is almost instantaneous and current starts to flow in a sub-nanosecond time after voltage is applied. You don't need to know Di/Dt unless the diode of interest is an DIAC or the anode and cathode of a TRIAC or SCR. Look at the data sheet(alldatasheet.com) for your diode to see switching voltage waveforms and forward voltages v. current. Most ordinary silicon diodes do not start to conduct current(microamps) until after approx .65V of voltage has been applied(room temperature) and the voltage rises up to 1.2V to 1.5V approx when the current approaches the maximum allowable for the diode. At higher temperatures, the threshold forward voltage of conduction will decrease with temperature at a rate of approx. -2.2mv/deg C of temperature difference from room temp.
What is often more important is the type of diode. Schotkey diodes will conduct with faster turn on and faster turn off and have a forward threshold voltage of .12V to .18V with a milli-amp of current noticed, depending on the diode.