Why the heck did this paper cut take over a full minute to start bleeding. It just seems really weird to me.
Capillaries are lined with smooth muscle, and under stress that tends to constrict. Damaged cells release chemicals into the area around them and stimulate nearby cells. Your body has defenses against bleeding. Most capilaries are barely wide enough for a single red blood cell to pass through them at any point, so even a slight constriction is adequate to seal the passage from blood flow.
Had enough of those chemicals been released, the capillaries might have stayed closed until platelets (which are essentially fragments of red blood cells bonded to a few additional proteins, and are much smaller than an intact red blood cell thus able to circulate and reach the wound despite the narrowing of the blood vessel) managed to form a clot at the air-blood interface. If that had happened, you wouldn't have bled at all.
Had the paper cut been deeper, it might have severed some blood vessels larger than capillaries. Those would also constrict, but probably not enough to prevent bleeding - in that case you likely would have started bleeding right away.
If your papercut was shallow enough, it could have missed even the capillaries, and then you wouldn't have bled at all.
In your case, most likely your paper cut damaged enough cells to release about a minute's worth of irritating chemicals, but missed any larger blood vessels. Once your capillaries' smooth muscle fibers got over the shock, they relaxed again and out came the blood.