Dear Jill!
Örülünk a levelednek. Milyen szépen tudsz magyarul!
Hopefully second time round you will rediscover Hungary again as a place where you can have great private healthcare options among other things. 11 years is a long time!
Telki is now closed down completely.
However, for private clinic I am aware of two options: Maternity Clinic and Robert Karoly Clinic.
I have given birth at Maternity in February (it was amazing), know most doctors there, while a good family friend of ours is the founding owner and chief at Robert Karoly so I guess I know a lot about both in terms of their staff which is kind of the most important bit (the facilities are great for both as these are private clinics). I do not think that either place will be a problem in terms of natural or medicated birth- however if you make up your mind I can help you with that, too- some midwives and doctors are great for natural and some are excellent with C-sections or epidurals.
There is also a place called Dr. Rose where you can give birth too but I have not heard anyone of the medical profession recommend it.
Have a look at the websites of both institutions and I am happy to help you with any insider info to help you decide :)
Julia