How long would it take for braces to correct a cross bite?

Depends. Is it dental, skeletal, or both? Is it single tooth or entire segment? Is resolving the crossbite fully corrective of your malocclusion or are there other problems to be resolved? Please see a qualified Orthodontic Specialist for evaluation. Research shows Orthodontists resolve malocclusions faster and better than non-specialists.
Varies greatly. There is no magic number and it depends greatly on your case. You would need to see an orthodontist to get a better idea of treatment length.
Cant be predicted. Without and evaluation. Have your orthodontist answer your questions.
It Varies. Please see an orthodontist so they can determine the severity of the cross bite and how many teeth are involved. Is only one tooth in cross bite or are there multiple teeth involved along with a skeletal component?
Depends on severity. Braces treatment might correct a dental crossbite in only a few months; however, it depends upon the severity and nature of that crossbite. Simple, single tooth crossbites are much different from those involving multiple teeth or narrow upper jaws. An appointment with an orthodontic specialist is the best way to determine if aligners will correct yours.
Exam needed. Cross bites can be divided into two basic groups: skeletal and dental. Dental crossbites can be more easily addressed, depending on the extent of the problem; skeletal crossbites are by nature much more difficult, and may require surgical as well as orthodontic involvement.