Then please seek training and support urgently around issues of safeguarding, especially as regards confidentiality, as you seem to believe it is safe and appropriate to share details related to specific children in your care with members of the general public not employed by or in your organisation. (As you think it was fine to share details of OP's son with all other parents without her consent).
OP, because of some of the random stuff that has been chucked in here (and I am sure you will be grateful all your days for the advice "tell your son not to bite" - if only you could have thought of it yourself, instead of sharpening his teeth and giving him a reward chart with stickers every time he draws blood, which is what i suspect you do....)
Sorry, my gallows humour. I am sure you are addressing his biting, and please be reassured, it's not an uncommon phase - I had friends whose kids went through it and, like everything, it does pass.
This is not a thread about biting, it's a thread about data breech. Can you tell us exactly how much identifying detail was shared, so we can help you maybe work out how best to respond and how far to take it.
For me, I would always want if possible to accept human error, so if no harm done and the nursery generally seems like a safe place from my child, I would potentially be willing to just tell them how I felt and seek reassurance it wouldn't happen again, and no further action.
But it does depend I think exactly what details were shared, and how, and why? (Was it an accident, definitely?)