The thinking behind the Children Act was that no one can properly say that they have “custody” of a child in the same way as you might have custody of a car, for example.
The focus is now on the relationship you have with your children rather than your authority over them so now both parents have “parental responsibility” for their children.
If you are (or were) married then both parents will automatically have parental responsibility for their children. No court orders are needed. Parental responsibility simply means that you should both consult over all the major decisions affecting the children e.g. change of school, change of religion, consent for non-urgent medical procedures etc.
As a parent with parental responsibility you can ask the children’s schools to keep you informed regarding open days, school reports etc. if your partner proves uncooperative on these issues.
You cannot change the child’s surname without agreement in writing from everyone who has parental responsibility.