It's actually a UK NHS hospital which operates a private maternity wing on the side. The email address in the brochure is even @nhs.net. They're probably relying on staff and resources from the NHS side to handle more difficult births too.
Private companies who provide services to the NHS also get @nhs.net email addresses. It's a secure email system so that patient and other confidential data can be exchanged between providers without it going out onto random internet mail servers.
An @nhs.net email doesn't mean "this person/organisation is a part of the NHS", it means they provide services to the NHS and need to deal with patient data.
As it says on the hospital's website, "difficult" births will be free to those eligible (British, EU etc) but will be charged to others (presumably people like the wealthy Arab people in the brochure).
