Recently I have had quite a number of patients with severe lip asymmetry from congenital cleft lip deformities. Your left side of your lip is deficient relative to the right. It is possible to increase the volume of your left side with a v to Y internal advancement flap. The lip could be lifted by removing some of the skin of the upper lip close to the nose. An examination by a qualified plastic surgeon is probably a good idea.
The problem you raise is a subtle but significant problem. Lifting the lip as part of a cleft lip revision is tricky to do well. I've done quite a lot of them. It would be best to have you evaluated in person in order to get the 3 dimensional shape of your lip. The recovery depends on the location of incisions made to do the lip revision. In general a scar takes about 8 months to fully mature, but women can use makeup to hide the scar after a few weeks. I would advise you to find a plastic surgeon with experience in cleft surgery. Good luck!