Standard vet training is on typical farm animals, horses, cats and dogs, with just a few weeks on other things that will be brought into the practice - budgies, snakes, hamsters, fish etc. They will probably have spent no more than a few hours on marmosets and squirrel monkeys, and none on apes. One who goes on to specialise in exotics - especially zoo work - will know more, but even places like Monkey World and London zoo bring in doctors for specialist work like ophthalmology on primates.
However, a key skill for vets is adaptability, and being able.to apply what you do know to an unfamiliar species. And the vast majority of drugs are the same (they'll have the BNF and MIMS available for working out human dosage). So in an emergency they'd have a far better chance of a decent result than an average person.
The cuts for a c-section might not be ideally positioned, but there's a very decent chance you and the baby would both live. In a vaginal birth there are complications in humans that don't arise - or are extremely rare - in other animals, so a vet might need to go for a c-section in circumstances a trained midwife could manage non surgically.
Post-birth prolapses look quite different in farm animals, so you might not want a vet treating one of those.
Stitching and placing IV lines would be no problem at all. Less fur and thinner skin makes it much easier on humans.
Re cows arses, I think the confusion may be between prenatal examination - which is sometimes done through the rectum to feel the uterus from the 'outside', and actual delivery which is definitely vaginal (unless it's a c-section, which goes in through the side).