I just finished a few cycles of the Cube, so let me give you my two cents.
On the heavy days, take as much time as you need. Since the weights are never really heavy until the final heavy day, I always tired to move them FAST. Now, if you are working hard and trying to blast these sets, you will need a little more rest. Take as much as you need. The point of this day is to get something moderately heavy on your back and slaughter it.
On explosive days, I took a different approach, basically trying to complete the work in as little time as possible so as to elicit the greatest training effect with the lighter weight. So, I do think you should move fast between your sets. However, I think there has been a break in communication; Brandon Lilly himself recommends 15 seconds because he pulls singles back to back with a partner, not triples. If you are doing triples, as I did, do them like any other speed sets, approx 45 seconds between sets.
Axle Press (270x4), Log Press (255x1), Farmer's (215/hand, 50 ft), Strict Press (225x1