I totally understand why (usually) mothers wish to change their children’s names to their own. Sometimes the name may be the only real link the children have with their father so the Court often thinks it is important that is retained. The Court may well agree to a double barrelled surname though.
The bottom line is that if Father will not consent to a change of name, the Court is unlikely to make an Order.
If Father cannot consent to the change of name (or is nowhere to be found) then an application for a specific issue order to change the name will be necessary and may possibly be successful.
Alternatively, School may accept a double barrelled name informally in some circumstances (although it will not be possible to do make changes on formal documents eg passport) or some clients of mine have actually circumvented the problem by changing or hyphenating their surname by deed poll to include the children’s surname.
However the most compelling reason NOT to change a child’s name is usually that mother’s name will change on marriage or remarriage (and however much mothers may not believe it a possibility at the time, it is more likely than not) so ultimately more change and confusion is inevitable.