The best answer is probably "It's magic, don't overthink it," but this is the PPC. Let's overthink it. {= )

So, the average gestation period for a human baby is 40 weeks, or 280 days.

The average gestation period for a foal is 48.5 weeks, or 340 days. That's, um, a lot. Almost a whole year.

However, a human baby and a foal have very different levels of physical development at birth. A human baby is completely helpless at birth; it needs to be carried in order to feed and to get around. A foal can stand up to suckle shortly after it's born, and it's able to run with its mother not long after that. Even if you let a human baby cook an extra couple of months, it will still be completely helpless, so simply using a mare's average gestational period won't help. We'll have to completely reimagine the entire process.

First, a centaur is a mash-up of a prey animal and an apex predator. Fortunately, both halves are social creatures. This is fortunate, because it means a centaur mother can rely on her family for protection while she and her baby are vulnerable. This means a centaur baby doesn't necessarily have to be at such a high level of development as a foal when it's born. It can be a little smaller, maybe small enough to be carried in an absolute emergency. This will make the birthing process easier, too, and we should take every opportunity we can to do that since we already have a lot of moving parts to worry about. (Ideally, a centaur baby would probably come out head first and arms back, like a human baby, and forefeet first, like a foal.)

So, the baby doesn't necessarily need to be able to walk when it's born. It does need to be able to feed without being carried, though, because that would just be absurdly awkward. Assuming her mammaries are located on her human torso, as most commonly depicted, Centaur!Mom probably feeds Baby while kneeling, and may use her arms to help hold Baby in the proper position if Baby needs help supporting its head and torso. But, I would guess that Baby can probably do this without too much trouble after, say, a few days to a week.

But, hold on. It normally takes a human baby about six months after birth to build up enough strength to hold up its head. Does that mean we're looking at an 78-plus-40 weeks pregnancy?

It could. It wouldn't even be totally unheard of: elephants have an average gestational period of about 94 weeks, or nearly two years. Yeah. Imagine that, if you dare.

You know what? Let's go with that. Give that extremely complex human brain as much time as possible to develop its ability to control its extremely complex body before birth. A two-year+ gestation should just about do that. The development of the horsey parts can probably be slowed down much more easily than the rest of the process can be sped up.

As to how this very long process will affect Mom physiologically and emotionally at various stages, I could not begin to guess. You could look up how pregnancy affects mares as a starting point, though. And maybe research elephants while you're at it.

... But seriously, the best answer is still "It's magic, don't overthink it." {; )

~Neshomeh