No, birth certificates cannot be changed, (although there are very limited and time restricted options to correct errors soon after birth). Errors on birth certificates are a common reason for changing names by deed poll.
If you were married when the child was born, fathers have a joint parental responsibility and must agree to the change. (If the father cannot be found the name can be changed but it must be done via a solicitor.)...
Provided no fraud or deception is involved and the name is not offensive, you can choose any name you want. We reserve the right to refuse to issue a Change of Name by Deed Poll if we consider the chosen...
Only if a solicitor certifies the copies. A cheaper option is to have a number of identically signed and witnessed Deed Polls. Overall, this is a quicker and cheaper option.