amberjane:
I'm so sorry to hear about your DD. She's got a lot on her plate and I'm sure you're worried sick.
I think you can't change the situation with her father, but you do need to let him know she's upset about it. He may do nothing, but at least the ball is in his court. I'd suggest you ask your DD to write to him or send a picture.
She's lonely. That's not instantly solved but some suggestions:
Join after school clubs or clubs at weekends. Swimming, ballet, language, etc...
If the school offers free violin or other music lessons say yes - this ultimately will lead to joining a youth orchestra.
Consider joining choir - usually free and very social.
The weight issue is difficult - it might be you are both eating a lot more than normal because of the situation. But possibly try and encourage healthy choices - fruit over crisps or carrot sticks over chocolate. I know it sounds dull - but getting into the habit is good for you both.
Try inviting some friends over. Possibly over the Christmas break. Ask your daughter who she thinks is nice in her class and send a note or approach the mother about having a play date over Christmas. Keep it calm and relaxed, don't make a big deal about it and that will help.
She will make friends - in a few months there will be another new girl or boy and odds are they'll become good friends. I know it's hard right now, but keep her busy and active and try and assure her that making friends takes time - those kids have had 3 years (Class R - Class 2) to be friends, possibly longer - so it is a slow and gradual thing. But she'll get there in the end.
In the meantime - there is nothing to stop you arranging for her to see some of her old school friends over Christmas. That might be a nice treat.