How are you reading the voltage ? I guess you must be using an DAQ card to do this right ?
If you have a card which can read the voltage in a specific interval of time (that you can specify) for a certain number of readings upon triggering (say every 10 microsec 1 reading for 10msec = 1000 ADC reading). The device must be having a buffer to keep all these data as well (so that you won't interrupt the device until it finishes). Once reading finished, you can read the buffer and look for the time reading 0.6 x applied voltage (or close). Once you see this reading, you can calculate the time as [the index of the reading x 10micro sec].
The key here is to find the card that can do it this way.