I think CC is too much too soon, if she's relying on milk and cuddles during the night. Try and see it from your DD's point of view. Mmmmmmmm, milk, cuddles with mummy and daddy in a nice warm bed..........
Oh FUCK, what is going on?????????? No milk, no cuddles, all alone in the dark, it's horrible!
Cut out the milk, but cuddle her to sleep beside her own bed, in an upright position (so it's vaguely uncomfy). lay her down when she's sleepy. During the night, let her into your bed.
As the weeks go by, if she visits you in the night, take her back but stay with her.
Play it by ear from thereon. If you still feel the need to do CC, it won't be so traumatic for her.
We did this for DS1, and DH still falls asleep with him, but he sleeps through the night.
I'm about to try it for the third time with DS2. He's awful, wails for hours.