When reading about St.Mungos I didn't see delivery room mentioned there. Do they go to Muggle hospitals? Do they deliver at home?
There's not any particular hospital mentioned in the book or movie.
However, in Order of Phoenix, there was a mention of Neonatal Intensive Care Unit in newspaper. It was an article written by Tilly Monk's husband. She gave birth to twins, Jack and Tom, there.
Other pregnancy cases were of Merope Gaunt, Lily Potter, and Nymphadora. Merope gave birth to Tom Riddle at the orphanage whereas there is no mention of rest.
Most likely they give birth at home.
In the absence of black magic, any normal ailments that can beset a pregnant woman would be trivially simple for a Wizard or Witch midwife to cure, so a hospital visit would be utterly unnecessary. We see at least one example of this on Pottermore. Neville Longbottom's mother was attended by a midwife.
Thus, the very moment that Neville Longbottom was born, the Quill attempted to write his name and was refused by the Book, which snapped shut. Even the midwife who attended Alice Longbottom had failed to notice that Neville managed to shift his blankets more snugly over himself moments after birth, assuming that his father had tucked the baby in more securely. Neville’s family persistently missed faint signs of magic in him and not until he was eight years old did either his disappointed great aunts and uncles, or the old stickler of a Book, accept that he was truly a wizard, when he survived a fall that should have killed him.