Will her UM apply to me, if we don't live together?
It might. The only way to know is to read the policy and see who it covers, and for what. There may also be legal decisions ("case law") that controls how the language in the insurance policy should be interpreted.
That's not legal advice as I don't hold Georgia licensure. If you need legal advice, please consult a lawyer who holds Georgia licensure.
Personal Injury Lawyer
Assuming that you were driving your mother's car with her permission, you are an insured under her policy and the UM coverage will apply and benefit you.
The accident should be reported to your mother's carrier. If you have been injured, I suggest that you contact a personal injury attorney before providing any type of statement to your mother's carrier.
Mr. Lundeen is licensed to practice law in Florida and Vermont. The response herein is not legal advice and does not create an attorney/client relationship. The response is in the form of legal education and is intended to provide general information about the matter within the question. Oftentimes the question does not include significant and important facts and timelines that, if known, could significantly change the reply and make it unsuitable. Mr. Lundeen strongly advises the questioner to confer with an attorney in your state in order to insure proper advice is received.