They'll do what they can, then get a cleaner onboard at the next stop, or mark the seat out if service until it can be properly cleaned. It's gross, but no grosser than any other human secretion that can be left on there.

If you bleed on an airplane seat, the airline treats it as a Biohazard/Bloodborne Pathogen incident. Flight attendants are trained to contain the area immediately. They will typically use a "Biohazard Kit" containing absorbent powder, disinfectant wipes, and gloves to clean the visible spill. However, because blood can seep into the seat cushion foam, the seat is often taken out of service for the remainder of the flight to protect other passengers from potential infection. Once the plane lands, a specialized cleaning crew (often wearing PPE) performs a deep decontamination. In many cases, the entire seat cover and sometimes the internal cushion must be completely replaced before the seat can be used again. If the bleeding was caused by a medical emergency, the crew will also file an Incident Report. You won't typically be charged for the cleanup, but the airline takes it very seriously due to health regulations regarding communicable diseases.