What would happen if a person ate a BB gun bullet?

The answer can be simple or very complex.

If the BB just struck the skin of the neck, you may have some bruising and pain.

If the BB went through the skin, there may be other complications: infection from a foreign body is possible. Also, damage to other structures is a concern. The neck has several important structures in close proximity - the trachea, carotid arteries, and spinal cord/vertebral column are all within about a hands-breadth of each other. A BB piercing through any of these structures can be life threatening and you should seek care immediately.