I've posted on your other thread regarding adoption but I'll just add a comment here about name changes.
As has been stated, anyone in the UK can use any name they like, at any time, as long as there is no intent to defraud anyone by doing so. When I was teaching many pupils used names which were different to those on their birth certificates. (There were several children in every class doing so). However it caused no end of problems when it came to those pupils sitting public examinations (GCSEs). Only the pupil's 'official' name can be used for examination entries. That meant that someone who'd been called 'James Parker' since he was a toddler (and might even have been unaware that his birth certificate named him as 'James Kersey') could only receive a certificate with 'James Kersey' on it.
There's no official way of changing a child's name but a letter from a responsible person, such as your GP, will usually suffice as evidence that the name has been changed. If I understand your post correctly, you've obtained such a letter. Please allow me to stress how important it is that you don't lose that letter. It could be vital to your daughter in the future, when she starts to encounter 'officialdom'.
Chris