All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Undergoing orthodontic treatment is permissible if it is intended to correct a defect or ward off harm like pain and the like. As for undergoing orthodontic treatment merely for cosmetic purposes and enhancement of one’s beauty, it is impermissible. You have to be certain or most likely believe that you need this orthodontic treatment and that it falls into the category of what is permitted by the Sharee‘ah. This is achieved by asking the orthodontist and acting upon his advice. Many scholars held that it is permissible to act upon the medical advice of a non-Muslim doctor if he is trustworthy. However, it is better if you can consult a trusted Muslim orthodontist. Otherwise, you may act upon the advice of the non-Muslim orthodontist if he is trustworthy, and you are not obliged to ask the same orthodontist who installed your previous braces. Rather, you should ask the orthodontist who will install the new braces because what happened in the past is over. As long as the orthodontist informed you that you needed to wear the braces, then there is nothing wrong in acting upon his advice, particularly if you cannot find a Muslim orthodontist.
Allah Knows best.