It's quite rare to be able to change a child's surname, if the father does not give consent.
Rare circumstances are those such as where a father is in prison, has been convicted of a crime against the child or there are other safety reasons to conceal the identity from the father.
Other than this very short list of exemptions, the norm is for the child to retain the name of their birth.
The compromise in many situations is to ask for a double barrelled surname to reflect both parents' names.
Of course your child's name can be changed if the father consents.
It would be a specific issue order application, on form c100 - here:
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