Unfortunately, I don't believe that there is anything you can do to overcome this limitation. Power is calculated as torque times angular velocity, with the former averaged over a complete pedal revolution and the latter based on the time required for that revolution to be completed. Thus, even if the crank is "awake" when you begin your effort, you won't get a power value until the end of the next full pedal revolution...as you've seen, this takes a while, especially if you're using a big gear (and have a puny maximal power, like me!

). The speed, OTOH, is recorded as increasing sooner, because the magnet on the wheel passes the reed switch on the fork more frequently (unless, of course, for some reason you're doing standing starts in a gear ratio of <1:1, e.g., on a mountain bike).
There is one trick I can pass along, which is this: if you set the recording interval to be significantly shorter than the time required to complete each pedal revolution (e.g., at 0.1 s intervals), you'll actually end up recording the data for each revolution, not simply as a function of time. This is so because the SRM always records the calculated power for full pedal revolutions...so if another full revolution hasn't occured since the last time data were written to memory, it will write the same data again. Sometimes using this trick can be of interest/value, although it will tend to fill up the memory very quickly.