Kido told parents the breach happened when criminals accessed their data hosted by a software service called Famly.
The software is widely used by other nurseries and childcare organisations, and it says on its website it is used by more than one million “owners, managers, practitioners and families”.
There is no indication other Famly customers have been affected. BBC News has contacted the firm for comment.
The criminals’ site contains a gallery of 10 children with their nursery pictures, date of births, birthplace and details – such as who they live with and contact details.
Parents have contacted the BBC concerned about the hack, with one mother receiving a threatening phone call from the criminals.
The woman, who did not want to be named, says she received a phone call from the hackers who said they would post her child’s information online unless she put pressure on Kido to pay a ransom.
The mother described the call as “threatening”.
Another parent, Stephen Gilbert, told the Today programme on BBC Radio 4 that someone in his parent’s WhatsApp group also received a call.
“The revelation the children’s details could have been put on the dark web, that’s very concerning and alarming for me.”