“What happens if you leave a broken jaw untreated?”

It is not pretty. It depends how long ago it was broken. To surgically correct a fracture that has sat to long, surgeons will have to re-break it. If it has been a month already, personally I'd wait and get insurance in January. Right now I would be leery of major procedures during the pandemic.

It can lead to the angle of your bite being imbalanced leading to chronic headaches and pain. Also, depending how bad it is, it can cause infection from contact with saliva. You do know that your mouth is the filthiest, most bacteria filled part of you body? Very bad idea not to have it set or looked after. Too poor to see a doctor? Many public health clinics and dental schools have very inexpensive charges to diagnose and treat patients. At least see a health professional. You might have to live with the consequences of neglecting this for a long time.

Due to constant chewing muscle activity, the fragments will constantly move and may never heal and will eventually cause osteomyelitis, especially when the fracture runs through a tooth socket. Some fractures may heal, but will cause a deformity in the jaw and a change in the bite.