How long does it normally take for a broken neck to heal completely?

This is dependent on. A number of factors including the age and health of the patient, the type and extent of the injury and how that injury was treated. The healing process can range from 6 weeks for simple spinous process avulsion fractures to several months for a type 2 odontoid fracture treated non operatively in an elderly person with a risk of it not healing.
It varies widely. The length of time can vary widely depending on age of the patient, severity of the fracture, stability of the fracture, associated injuries, associated medical conditions and patient nutritional status. Diabetics and smokers, among others, heal slower than those without these risk factors. You should see a specialist and seek their advice regarding each individualized scenario.