No canon answer. Also it is very hard to make assumption for the actual state since the capabilities of healers are quite contradictory when it comes to healing "muggle" diseases and injuries.
Here some examples:
Healers seem to be quite capable of curing viral infections like the Common Cold:
Harry Potter and the Chamber of Secrets chapter 8
October arrived, spreading a damp chill over the grounds and into the
castle. Madam Pomfrey, the nurse, was kept busy by a sudden spate of
colds among the staff and students. Her Pepperup potion worked
instantly, though it left the drinker smoking at the ears for several
hours afterward.
Instantly curing the Common Cold is something that muggle medicine has not achieved yet.
Injuries are also healed quite fast. However some restoration procedures include enduring unpreventable pain:
Harry Potter and the Chamber of Secrets chapter 10
"You should have come straight to me!" she raged, holding up the sad,
limp remainder of what, half an hour before, had been a working arm.
"I can mend bones in a second - but growing them back"
"You will be able to, won't you?" said Harry desperately.
"I'll be able to, certainly, but it will be painful," said Madam
Pomfrey grimly, throwing Harry a pair of pajamas. "You'll have to stay
the night ......
On the other side healers are not capable of healing some chronic diseases.
Examples include:
- Myopia - Harry, Dumbledore, McGonagall wear glasses. This is also a point for the muggle doctors since Myopia is curable for some time now.
- Lumbago - Perkins had to give up on camping
- Hearing loss
Harry Potter and the Order of the Phoenix chapter 22
A very old, stooped wizard with a hearing trumpet had shuffled to the
front of the queue now. 'I'm here to see Broderick Bode!' he wheezed.
Healers were also not capable of curing the psychological damage that the muggle boys caused on Ariana Dumbledore. Whatever these boys did it wasn't of magical nature. This is another point for muggle doctors since there are special therapies for kids that went through trauma.
In regard to the second part of the question:
Finally, are there wizarding midwives, or do people go to St. Mungos when they're about to give birth?
There doesn't seem to be a special section in St. Mungos for women that are about to deliver:
As the wizard hobbled and pranced sideways out of the way, the Weasley
party moved forward a few steps and Harry read the floor guide:
ARTEFACT ACCIDENTS...................................... Gound floor
Cauldron explosion, wand backfiring, broom
crashes, etc.
CREATURE-INDUCED INJURIES........................ First floor
Bites, stings, burns, embedded spines, etc.
MAGICAL
BUGS.................................................... Second floor
Contagious maladies, e.g. dragon pox,
vanishing sickness, scrofungulus, etc.
POTION AND PLANT POISONING...................... Third floor
Rashes, regurgitation, uncontrollable
giggling, etc.
SPELL
DAMAGE..................................................... Fourth
floor
Unliftable jinxes, hexes, incorrectly applied charms, etc.
VISITORS' TEAROOM / HOSPITAL SHOP.......... Fifth floor
IF YOU ARE UNSURE WHERE TO GO, INCAPABLE OF NORMAL SPEECH OR UNABLE TO
REMEMBER WHY YOU ARE HERE, OUR WELCOME WITCH WILL BE PLEASED TO HELP.
There is no special delivery section and it will be quite unwise to place pregnant women among sick/injured people.
So either the women deliver at home, in a specialized hospital or even in a muggle hospital.