You're 15 and this girl is 14. Your grandmother is afraid of you guys getting too close, becoming sexually active, and whatever else that may follow.
You've hidden your relationship with this girl, essentially lying about this and were caught. Your gran asked about you having sex with this girl - you said no - she didn't believe you as you had lied to her previously. The girl's mum asked you to back off, as she - as an adult, wanted to be respectful of your gran's request.
The thing is - once you've lied to someone you love, it's hard to regain their trust. You must go to your gran, and your mother, if you've lied to her as well, and apologize for your behavior. Sit down, facing each other, knee to knee and take her hand and tell her that you're sorry for lying to her. Tell her that you love her.
You're not doing this so you can date this girl - you're doing this because it's the right thing to do, so they will learn to trust you again. Speak from your heart and tell her how you were feeling when you lied and how you felt you couldn't be honest with you about this.
Tell her that you realize that this was a mistake and that you have learned that lying doesn't solve anything. You need to heal this relationship before trust can be restored. It could be that after a suitable time, she will like this girl and you guys can date - but you have some work to do before that happens. Good luck.