Anxiety is a real bummer. I understand. I think that your mom has laid down the law, and it is a real worry that if you contact your friend or get together with them that you will be in trouble/ have to 'hear it' and deal with it is likely. And then if you neglect your friend, it would jeopardize the friendship is also a worry. These two worries are in conflict of each other causing anxiety. This is not your anxiety at play but legitimate. Who wants to blow off a friend for their preference of mates that someone else doesn't approve of? That is really backward thinking on your mom's part. Is this the only reason she is having an issue with the friend? I know as a mom myself that I do look at my kids friends and judge things that feel 'unsafe' to me. If it is a kid who gets in trouble a lot, has a foul mouth, is mean to others . . . I have a problem with it and don't encourage a friendship. I think that is natural for a parent to do. But I try to keep it all in check too. My kids have to make their own choices. But it is sad that someone has an issue with someone simply because they choose the same sex. Is the worry that you will too? Ugh. Very small thinking there.
But you can't really change that I suppose so have to come up with a plan.
A. talk to your mom. If it is simply because your friend is gay, ask her why that means the person is not someone to be friends with. What it has to do with you? What is her fear or concern? And then tell her that you care about this person and it hurts you that she wont' let you be friends. See if she is will to bend.
B. if she won't bend on it, then talk to your friend (after telling your mom that you are going to have one conversation with your friend to explain, like it or not-- which she may resist as it will mean her bigotry will be exposed but this is the road she is leading you down, so so be it). in this conversation, explain that you care for them, wish it wasn't like this but your mom is making a stand. And you will be in trouble if you do not do as she says. And that you want to remain friends but it is loosely at school or out and about in groups until your mom sees the light.
At least that is what I'd do. But there is no good answer to this. I'm sorry about the situation. It's a hard one.