The biological parents remain the legal parents. Being a step-parent does not confer any particular legal rights so if you wish to become a parent, you must formally adopt the child. If you wish to have parental responsibility, you will have to be granted this either by the mother in the form prescribed or by order of the Court.
A civil partner with parental responsibility does not automatically become a parent. The civil partner of a natural parent is the child's step-parent. Having parental responsibility does not in itself make you a parent. In order for the civil partner to become a parent, they would have to adopt the child in accordance with the specified procedure.