Removing ex-partner from joint ESA claim

Good afternoon,
I am looking for advice and would really appreciate any help. 

My ex partner is in the process of moving out to her own flat, she picked up her keys on the 23rd of September but is still staying with me until all her furniture and appliances have been delivered to her new flat which should be a week today.

Yesterday I called ESA (legacy ESA) to take my ex off our joint claim so she can make her own single claim for UC but I was unsure what date to tell them for her moving out date so the person I spoke to was not able to take my ex off my claim. 

Should I give them the date she picked up her keys or the date she will be staying there from? and can she start the process of claiming UC even if she has not been taken off our joint claim yet?

Thank you