Mediation sounds like the best way to go - your dd has had a lot happen in her life - you and her father splitting up, you starting a new relationship, and then facing the move to a new area and new school - that is a lot for a teenager to cope with, alongside all the normal teenage angst and hormonal upheaval.
For the moment, she is in a safe place, with people who love her - it could be a whole lot worse - she could have run away and not been found or she might not have loving relatives willing to look after her whilst this is all worked out. Is your mum that bad that staying with her is worse than your dd being with strangers or on the street?
You need to give her some space, and you need to try to see things from her point of view. Maybe she felt that everything in her life was out of her control - whether her dad and mum were together, her mum's new relationship, and the move was the last straw - so by running away, staying with her grandparents, and refusing all contact with you, she is taking back some control.
However, I don't blame you for being cross with your mum, because it sounds as if she is making the situation much worse than it needs to be. If you were my dd, I would be supporting both you and your dd - my granddaughter would be welcome to stay, but I would be trying to maintain the channels of communication between the two of you.