There îs minimal visible swelling from alar rim grafts alone. There would just be swelling along the edge of the nostril and the nostrils may appear a bit wider from the front view. Usually the cartilage is taken from your septum so you will have some nasal congestion. You should be socially acceptable really in a few days if this is the only procedure you are having done. If however you are having a complete rhinoplasty then 2 weeks is more realistic. I hope this helps and good luck with your surgery.
Initial visible swelling is a few weeks after placement of the grafts, however full healing and integration of the grafts into the tip of the nose will be one year.