The earliest that this surgery can be done is after the age of 18 with parental consent. So mom is quite correct that way ! However ideally speaking this procedure is undertaken in most women once they have completed their families as it might interfere with breast feeding in some cases. So unless you are having severe neck discomfort, have massive breasts giving you a very poor quality of life, its better to wait a bit.
You have not provided information regarding your height, weight, bra size, etc. However, in general your mom is right, but there are situations where we have no choice but to perform a breast reduction at an earlier age. There are girls who at your age are around 5'5" in height and weigh 120 lb, yet their breasts are a DDD or even F cup. In such cases, yes surgery can be performed before age 18, but they have to be prepared to undergo another possible operation in the future because they stand a high chance of regrowth of the breasts.
Insurance requires you to be over 18 and it is also advisable to insure that you are fully developed. In the meantime, you can pursue physical therapy for back pain. Remember that you can't have any surgical procedure before age 18 without parental consent,
If you are 16 and your mother thinks that you need to wait until you are 18, then that is your answer. The surgery has been completed safely and effectively on patients that are your age, but they have the support and consent of their parents to do so. If your reasons for having the surgery are mostly cosmetic, then I would tend to agree with your mom. I think that cosmetic surgery for patients under the age of 18 is not usually the best idea. But if you are having serious back and neck pain symptoms that are causing a medical issue, then bringing your mom to a consultation with a plastic surgeon might open her eyes to the medical benefits that the surgery can provide.