As already said, I believe it depends on your obsteotrician for private hospitals like the Matilda. If he has "a big face" at the hospital, or lives there, your chances are probably higher. We had no problem at 9 weeks to get a confirmation, but we know from other people at Geobaby that they were not so lucky (and I can now confirm they are very busy these days)!
Somehow I also have the feeling that if you select an expensive option (C-section and private or VIP room), your chances are higher, but this is just speculation.
The earliest your doctor can send the booking is 9 weeks and with blood work results and ultrasound.
For us it only took around a week to get confirmation from the hospital, but I've heard others say it took a month to get rejected. If that is true, that is indeed worrying.