There is a method called "6 Month Ortho" that was developed for JUST this purpose. The average case takes around 4-8 months. If you aren't trying to change the bite and a few other limitations, this may be the best choice for you.
I would recommend getting the rhinoplasty first ONLY if there is nOT going to be much movement in your teeth. If you have a big overjet or are having teeth extracted from the upper arch, then i would wait until after the braces to do the rhinoplasty.If you are going to do the rhinoplsty first,...
Moderate gaps of the anterior teeth close rapidly. It may take about three to six months to close a moderate space. The time of closure of teeth spacing also depends on the age of the patient and a few other factors.