When I was teaching nearly one third of all pupils in the school used a different surname to the one on their birth certificate. It's common practice.
However the girl's father has parental responsibility and his permission is required for a 'legal' change in her name. Unlike in many other countries though, in the UK anyone is free to use whatever name they (or their parents) choose for everyday use (as long, of course, as there's no intent to defraud someone by so doing). The girl's mother can, irrespective of the father's permission, call her daughter by whatever name she chooses.
It might be worth advising the girl's mother that unless some form of 'evidence' of a name change can be produced (such as a letter from her GP), the girl will have to be entered for any public examinations she might later take (such as GCSEs) under the name on her birth certificate, which could result in her obtaining certificates in a name other than the one by which she's known.
Relevant links:
http://www.direct.gov...entsrights/dg_4002954
and
http://www.adviceguid...ly/change_of_name.htm
Chris