How long until anemia can cause some serious harm?

Depends on cause... There are many causes of anemia, and each will cause a different degree of anemia and will be different in how quickly they drop a person's red blood cell count. In an extreme case, say severe autoimmune mediated hemolytic anemia, the person could die with a couple of days. On the other side, many causes of anemia result in stable disease (does not progressively get worse).