How long is the recovery time after a facial fat transfer? I can't take too much time off of work for surgery, but I don't want to go into work with a swollen, unnatural-looking face after having a fat transfer to treat my wrinkles. How long will it take

Probably too long. Facial fat transfer to restore volume loss with aging that will work will leave you swollen for several weeks. Patients frequently go back to work after two, but can remain swollen for several depending on the volume, and health of the patient. If you are in a rush for (any) results after (any) cosmetic surgery (except possibly synthetic fillers, you will most likely be frustrated.
Fat grafting. It all depends on how much grafting. You will definitely have bruising and swelling. This will improve ovr time.
Check with surgeon. I would check with your plastic surgeon. They will have a good idea of the swelling and bruising. There is a supplement that some of my colleagues have used called arnica montana which they feel reduces swelling and bruising significantly after surgery. Check with your doctor first. You never never want to take anything your surgeon does not know about.