In the Netherlands, when a father acknowledges their child (in my case 18 months after my birth), they can change the name immediately upon acknowledgement to his surname. That's what he did. The birth certificate gets amended and that amendment shows the name change.
The Dutch government states at https://www.government.nl/topics/registering-a-birth-and-name-of-child/question-and-answer/which-surname-can-i-choose-for-my-child:
The birth certificate is evidence for the name change, given that it is amended, so does that mean I'd need to upload it at the place where I uploaded the marriage certificate? Or should I still wait for the NVC stage?