What is the longest recovery period for a broken bone?

Varies. We in medicine frequently see non-union fractures, or fractures that just never heal. It depends on the type of fracture, your unique medical challenges, age, and usage. Most fractures have formed a good bone callus in 6-8 weeks. Healing then proceeds over the next 4-6 months with bone remodeling, in most typical limb fractures.
Typically 3 months. Fractures typically take around 3 months to heal; however, depending on the specific bone and circumstances around the fracture, recovery can sometimes take longer.