My feeling would be that a Key Worker shouldn't be in a relationship with a parent, who's child is assigned to them. It's a conflict of interests. Working at the nursery, that your child attends, shouldn't be an issue, but she shouldn't be your child's key worker.
I work in a school, and we had a class teacher who had a child at the school, she wasn't permitted to teach her child's class. It's exactly the same thing really. My previous school I worked at, is the same one as my son attended. I didn't deal with any incidents involving my son, they were all passed to another member of staff.
The nursery should remove your child from having this young woman as their Key Worker, and should assign someone else to them. I don't think that would be an unreasonable request.
The taking your child off site without your consent or your ex's, is a different matter. You need to find out from the nursery what you're signed, and why it happened. It could be the young woman, thought it was OK because she's dating your child's Dad. It seems an unlikely explanation though, as she'd have to be completely clueless and rather stupid, to do such a thing. I suspect there's been a communication error or the young woman made a genuine mistake.