Was it because of gangrene or infection? I had to have the end of my iguana's tail amputated shortly after I adopted him, because it had dry gangrene. This is a condition where the tail is injured and becomes infected- it shrivels up and dies, and can spread from the tail tip to the entire body. It can happen because the iguana whips its tail a lot and injures the whippy end part, or if the tail is stepped on (my iguana's former owners 
If gangrene is why the former owners had his tail clipped, then you will want to watch the regrowth very carefully to make sure they caught all the gangrene. When your iguana's tail is regrowing, it will start out as a smooth black or dark brown bump, and then grow longer. It may or may not regain some of its coloring. It will probably never look the same as the rest of the tail, but if your iguana is young enough, it might grow back to its usual length. It will take a while longer for it to regrow scales of any type.
If the tail is not growing back, or more of the tail above the cut part becomes dried up and dark, or oozes pus or becomes mushy, take him to the vet right away for a second amputation.