Hmmm ... this is right up there with "what happens when the skater in front of you barfs during their program". With vomit, usually, scraped off and then a zamboni ... but, for blood ON the ice, it's usually just - there - until the next flood. Where it drips around the rink, then, the whole bio-hazard blood-absorbing powder plus swabbed down with some kind of antiseptic.
My DD had to follow her friend on the ice after the friend cut herself on a catch foot spiral (note: skaters need to practice anything that involves catching the blade BEFORE competitions WITHOUT gloves so they know whether they are catching safely) - she said she was so worried about her friend, and was watching to make sure she didn't skate into the blood spots, that she had a fantastic skate; it distracted her from any type of nerves ...