If you want a new birth certificate you’ll have to order one separately - you don’t get one automatically when you re-register the birth.
If the wrong father is named on the current birth certificate, you can ask for a correction. That will remove the wrong father’s name. Then you can re-register the birth with the right father’s name.
If there’s been a ‘Declaration of Parentage’
A Declaration of Parentage is when the court decides who the father of a baby is. You don’t need to do anything to change the birth record if there’s been a Declaration of Parentage made by a court. The court will tell the General Register Office and the birth will be re-registered. You’ll need to order a new birth certificate if you want one.