You don't give much information on how the needle got stuck but I'll assume it is not stuck by dropping the airbrush on the head. And I'll assume you didn't push the needle in the head so far that it has a mushroomed the needle. That would make the nozzle act like a hook and you would not get the needle and nozzle apart without damage.
You didn't say what you used to spray the airbrush with. Acrylic or enamel. If it was acrylic then a bath with ammonia and soap for an overnight soak could work to soften your clog. If it was enamel you could try to soak with either thinner, which is harsh, or vegetable oil, that is so soothing to the airbrush, which will soften the paint enough to pry the needle and nozzle apart after an overnight soak.
Try to soak the tip of the airbrush only. And twist the needle as you pull on it.
You can always remove the stuck parts with the needle wrench as one piece and replace the parts.
Good luck!