The mother has the final say on the surname at registration as she is the only one who has parental responsibility before that document is signed.
If you want a change of surname you need to apply with a specific issues order. If you have no current arenagments in place I'd advise doing that with a c100 as a specific issues order without a child arenagments order would look like you are more focused on the surname over seeing your children.
Highly unlikely you can get it changed to your surname only unless you can show it's it's children's best interests not to have mothers surname. The threshold is rather high for this.
Double barrell surname request are much more common as it is very hard for mother to argue why it's not in best interests of child to not have fathers surname in part.
She will likely use the "primary carer" argument, but that's equally shit down with the opposite. In that if the children live with mother than it's more important that connection is maintained if they are having les time with you.
The primary case law that is referred to is Dawson Vs wearmouth. If you have a Google around, you will see that any surname change has to have regard for the welfare checklist.