Full applicant's name on child applications
You must fill in the full name of the person who will execute the deed poll (sign it in front of a witness), and this person should be someone who has parental responsibility for the child.
If you (the applicant) are also in the process of changing your own name, you should probably enter your old name, because your new name will not be legally recognised until you have updated your official records. The name you sign the child's deed poll in should be your legal name at that moment.
If you are changing your own name and you want to sign the child's deed poll in your new name, you will have to wait until a significant number of your official records have been updated before signing (and certainly your primary identification documents — that is, your passport, driving licence and any ID cards).