Legally (in the UK), you can give a child ANY surname. (It doesn't have to be that of either the mother or father).
Where a couple are living together 'as man and wife',and expect to remain in that relationship, it would probably be logical to use the man's surname (as, in the vast majority of cases, that's what would happen if the couple were actually married).
But where the couple aren't living together (or where the relationship isn't 'secure') it would make more sense to use the mother's surname.
As a former teacher, I'd advise that you need to remember that a school is obliged to enter a child for their GCSE examinations in their 'official' name (which is the name which will appear on their examination certificates). I remember countless problems where (for example, and using fictional names) a young person who had always been known as 'Paul Smith', and who had never even met his father (Steve Jones) found that his exam certificates were going to be in the name of 'Paul Jones', simply because that was the name on his birth certificate.
Chris