It looks like specialized skills are needed for magical healing. Even Dumbledore would take injured folks to Poppy Pomfrey.
“Get Madam Pomfrey,” whispered Dumbledore, and Professor McGonagall hurried past the end of Harry’s bed out of sight. Harry lay quite still, pretending to be asleep. He heard urgent voices, and then Professor McGonagall swept back into view, closely followed by Madam Pomfrey, who was pulling a cardigan on over her nightdress. He heard a sharp intake of breath.
“What happened?” Madam Pomfrey whispered to Dumbledore, bending over the statue on the bed.
“Another attack,” said Dumbledore. “Minerva found him on the stairs.”
Harry Potter and The Chamber of Secrets, Chapter 10, The Rogue Bludger
Not all issues can be solved by wand work.
Hermione and Madam Pomfrey came around the curtain. Madam Pomfrey was holding a large bottle of something labeled Skele-Gro.
“You’re in for a rough night,” she said, pouring out a steaming beakerful and handing it to him. “Regrowing bones is a nasty business.”
Harry Potter and The Chamber of Secrets, Chapter 10, The Rogue Bludger
Some injuries can be fixed in minutes
“How’s your arm?” said Harry.
“Fine,” said Neville, showing them. “Madam Pomfrey mended it in about a minute.”
Harry Potter and the Sorcerer's Stone, Chapter 9, The Midnight Duel
But based on the injury type, it can take anywhere from few minutes to many days of rest for injured people to recovery.
Some take days:
“How long have I been in here?”
“Three days. Mr. Ronald Weasley and Miss Granger will be most relieved you have come round, they have been extremely worried.”
Harry Potter and the Sorcerer's Stone, Chapter 17, The Man With Two Faces
But generally, magical injuries take longer to recover from. Mr. Weasley spent days in St Mungo's Hospital for Magical Maladies and Injuries because his bleeding would not stop.
“Why can't they take them off, Dad?" asked Fred.
"Well, I start bleeding like mad every time they try," said Mr Weasley cheerfully, reaching across for his wand, which lay on his bedside cabinet, and waving it so that six extra chairs appeared at his bedside to seat them all. "It seems there was some rather unusual kind of poison in that snake's fangs that keeps wounds open. They're sure they'll find an antidote, though; they say they've had much worse cases than mine, and in the meantime I just have to keep taking a Blood-Replenishing Potion every hour.”
Harry Potter and the Order of the Phoenix Stone, Chapter 22, St Mungo's Hospital for Magical Maladies and Injuries
Since lots of injuries caused during the Battle of Hogwarts would have been magically induced, it is hard to say how many people could have been saved. But I don't think Medical help were lacking. It's just that too many folks were getting injured.
Now, we have time for one video!
The famous Skele Gro scene:
