Baby Birth-hospital expenses-Hungarian Medical insurance

Even if she is a Hungarian citizen  and  enrolls in the National Health, she will not be able to use it for the first year unless it is an emergency situation.
I think you will have to look into private care since you don't have a year to wait.
We moved back to HU and my husband is a citizen. He had to enroll again pay monthly for national health but couldn't use it for the first year. The second year they didn't bill him and on the 3rd year was billed monthly for coverage.
I have forgotten exactly what my husband read once about tips prices for doctors for a birth. I know they were high, over $500. to $1,000 for tips.
Not sure what they expect in a private clinic as far as tipping goes.
There was a big law suit brought by a few new parents for a couple of doctors and their staff. They were using a strong arm tactic to scare new parents into giving large tips. The nurses actually spelled out how much they wanted and how much each person should get in way of tips for their services.
Not cool to scare new parents that way. Not sure what happened in court, that was an extreme few cases.
In way, if you  pay out of pocket for delivery, it will be allot cheaper then in the US.
I can't recommend any one private health insurance co. sorry on that point.

Thank you for this information Marilyn we decided to postpone traveling to Hungary at this time we will wait till we meet our baby    thanks a lot.

Good luck and best wishes.