I didn't tell anyone when I stopped drinking and that worked out well for me.
It's natural that your mother is concerned about you, and that she may be afraid of what will happen given the brother she had who ended up on the streets. I think the best thing you can do is to show your family that you are changing. When you see them, let them see positive changes in you. If the topic is now taboo (your mother's choice), then respect that and focus on your recovery.