10 dats. The latest data shows that a good immune response develops within about 10 days of receiving the vaccination. But for children under 10, they don't develop a strong immunity with just one dose. That's why they need to get a second vaccination within a month of the first one.
About 2 weeks. After a person receives a flu vaccine, his body starts to identify the virus (if live vaccine) or pieces of virus (if non-live vaccine) as foreign. Then his immune system, with its white blood cells, lymph nodes, and spleen, remembers the virus's parts and makes antibodies against them. In about 2 weeks, his body should be able to recognize and destroy the 3 flu virus types in the annual vaccine.