I have an earlier answer that looks at late medieval and early modern Europe as far as maternal mortality. I'll c-p and then expand on it here:

I can talk about England, and hopefully others will come along to compare. I'll give the estimated statistics, and then I'll discuss why we can trust them for the late Middle Ages.

Studies from the 1970s that looked at cities produced rates of 2-3%. Forbes and Wilson posited around 3% for London in the late 16th century, extrapolating from records from one hospital that put the mortality rate in live births around 2.3% and adjusting for the addition of stillbirth deaths. Dobbie saw 2.5% for Somerset. In 1986, Schofield's highly influential study reduced the number downward sharply, to 1-1.5%. Schofield brought two innovations. One, he studied rural parish (baptismal/funeral) records. Two, he found what are apparently considered better ways of calculating the impact of stillbirth occurrences on maternal mortality.

Scholfield had actually calculated a rate of 1% (or 10 in 1000) for the rural parishes, but it does seem to be a pattern that maternal mortality was higher in cities than in the country. Why? Cities were dirtier! So with a rate of 1% per birth for the countryside, the higher rate in the cities would raise the overall total a little bit.

Now, why can we trust these statistics for the late Middle Ages?

First, the numbers seem pretty constant over most of the early modern era. (There is some uptick in London in the late 17th century, I think because of increased pollution and dirt). They don't start to fall until rising literacy among midwives coincides with the dissemination of printed manuals of childbirth--midwives had better, more up-to-date training in what to do when things went wrong. So it seems fair to extrapolate backwards a century to ~1400.

Second, Schofield actually worked comparatively between England and Sweden, finding fairly consistent/matching mortality rates. If this is a European trend relating to education of midwives (surgeons were typically brought in only if something was already very seriously wrong and death was likely), as the falling rate towards the end of the early modern era seems to indicate, it was probably rather consistent over medieval Europe. Herlihy and Klapisch-Zuber calculate a maternal mortality rate of 1.15% for early 15th century Tuscany (Florence and surrounding towns/countryside).

As a final note, it's important to realize that this 1-1.5% mortality rate is per birth. Schofield asserts that the rate probably increased along with the mother's age at childbirth. Overall, scholars usually credit the average woman with 5 pregnancies (although I'm not entirely sure how this was calculated, sorry!), yielding the chances of an individual woman dying in childbirth over her lifetime at around 5-10%.

(ETA: I know, math, but 10% overall is the number frequently bantered about.)

I neglected to answer the follow-up question in that thread about birth order and potential for difficulties. Elsewhere, although I can't find the link right now, I've talked about folklore surrounding birth (songs and such) in early modern England. It seems to emphasize a particular danger around first births, although the study where I read that said the statistics don't particularly back up that phenomenon. One reason for this is of course age of mother at birth. Late medieval and early modern women typically married much later than we think, especially in northern/northwest Europe where these statistics tend to originate, so too-young mothers is much less a factor--but also, women giving birth to pregnancy 9, 10, or 11 might be fairly advanced in maternal-years.

In this Tuesday Trivia post from awhile back, I talk about another example of folk belief, in this case late medieval, surrounding childbirth and mothers' fear of dying: the quasi-magical use of the legend of St. Margaret of Antioch, dragon-slayer.

As far as child mortality goes, the greatest danger was within the first two months of life, due to birth complications (a lot of conditions treatable today in the hospital with early intervention would kill within days or weeks before modern medical care) and tapering off gradually. Overall, scholars calculate around a 50% mortality rate by age 14.

While maternal mortality rates in the West did not start to improve until the 19th century, really, I do not know whether the same is true for child mortality.