If a tooth has had to be extracted for orthodontic reasons, there is a healing period afterwards. How long it lasts can vary depending on the type and size of the tooth removed, the body's ability to heal, and whether there is any problem with contamination or inflammation of the wound. In general, it's advisable to fit an orthodontic appliance once the discomfort from the extraction has passed and the soft tissue has healed. For premolar extractions this means at least one week; after wisdom tooth removal it may be worth waiting more than two weeks.
If you go ahead with extractions prescribed by your orthodontist, it's best for treatment to start soon afterwards – no more than 3 months should pass between the extractions and fitting the appliance. Of course, it's possible to start treatment later, but you'll lose the chance of faster initial tooth movement that comes when the extraction is recent.
This answer is for general information. Your own case needs an examination by an orthodontist.