According to NHS Choices, blue fingernails are caused by low blood oxygen levels or poor circulation brought on by exposure to cold temperatures. This condition is called "peripheral cyanosis."
The Mayo Clinic explains that when the body becomes cold enough that the core temperature drops, the small blood vessels in the body constrict, restricting the flow of blood to the extremities. This reaction alone is not always enough to cause fingernails to turn blue. However, as hypothermia progresses, the heart rate slows, further limiting the amount of blood that is able to reach fingers and toes. By the time hypothermia reaches this point, most people experience a change in skin color in the extremities.