But the op posted at just before 6pm asking what to say at nursery tomorrow.
If the accident happened just before pick up, and is a school nursery (which was suggested in the OP), then the neighbour could have as designated picker-upper called the OP, taken the dd straight to A&E or taken her home. If its a school nursery, then the OP was itself a couple of hours after pick-up. If not, and a nursery with later hours, then the fact that the nursery didn't call the OP is only a problem if the injury happened long before the OP's daughter was released to the neighbour.
The OP seems to think that this is the case on the basis of her dd saying she watched a dvd after the injury. As previous posters have suggested, this might not have been a delay, but have been for 15mins for example between injury and pick up.
Of course the OP can query this with the nursery if she is concerned. But on the face of what the nursery have reported i.e. injury happened not long before they would have expected the OP or responsible adult to collect, in noting the incident and getting neighbour to sign accident book the nursery seems to have followed correct procedure.
I certainly know how horrible it is when dcs are injured. Have had some heart in mouth moments when they've been in my care and also when at school/preschool. It's horrid. But there doesn't necessarily have to be a fault or someone to blame. Accidents happen and of course the OP can clarify with the nursery the timings if she feels they are in dispute.