There is definitely a delay there, but it's not horrible, and once you get used to it, you'll be able to time it just right so that it's not noticable. I've also heard that it's a fairly easy fix, but I don't know how.

I think the SD-1 will sound great along with the amps overdrive. I've got the MXR Wylde Overdrive, which is very similar to the SD-1, and it sounds great. So, you should be fine and dandy with that the pedal.