Hi,
St Istvan was my back up hospital, I was planning for a home birth, but ended up there after 25 hours of labour due to a botched cervical stitch that wouldn't come out and had to give birth with all of the stitch around the cervix (though cut in one place). So yes, they can handle difficult cases and pretended not to notice that stitch.
The doctor upon my arrival was English speaking, she had worked in the US and was very young and sweet (though I did get a couple of jibes about the failed home birth). The midwife spoke no English, but was very good. I was however forced to give birth on my back, you can avoid this by hiring an OB who works at the hospital who also happens to speak English, Attila Ujvary is the one I had, but I didn't retain him for the birth, though he is excellent. There are others the Moms will be happy to recommend.
They do avoid c-sections and episiotomies. The labour room was large, sunny and bright, no tub though. I don't believe you can get a water birth there, but I did not ask for one.
The recovery rooms are typical Hungarian old style rooms with 4 Moms plus babies in each room and food is terrible, I relied on my husband for some good coffee and break fast each morning. By the last day, I wanted to run out of the place :)
So if you want a natural, least intervention kind of birth, this is the place for you, though I hated giving birth on my back, I had the worst back labour, on the other hand, my baby did wrap the cord around her neck, so the nurse had to push her back in and release the cord which was a lot easier with me on my back, so not all bad.
Get an OB from the hospital to work with you before hand and brace yourself for some bad food, LOL. I highly recommend this hospital though, I did go back to take a pic with the doc as a keepsake.