Now, before anyone gets all uppity, read this to the end.
If I really wanted to do this, I'd treat pregnancy and delivery as a disease / poison.
Once labor starts, the mother needs to start making Fortitude saves. I haven't picked a DC, yet, but it's probably pretty low. Each save represents one hour of labor. Each hour after the first, the DC increases by a certain amount (probably +1 or +2). The mother must make ... 3? ... saving throws in a row in order to be "cured" - which, in this case, means giving birth to a happy, healthy baby. A failed save does 1 point of temporary Constitution damage to the mother (which, as normal, heals at 1 point per day).
Per the rules on providing care, an attending healer or midwife may make Heal checks. The mother may apply the result of the Heal check or her own saving throw, whichever is better, each hour. The attending healer or midwife, if they have appropriate spells available, may also cast them during the delivery (lesser restoration might be of particular use).
I'd also recommend clerics of certain sects having access to hourse-ruled spells like "Fertility's Blessing" (Chauntea) or "Dawning Joy" (Lathander), which grant long-term, small bonuses to Fortitude saves (+2, +4, etc.) made to deliver a baby, and reduce the number of successes required to deliver to 2.
By doing so, you reinforce the idea that having certain clerics on hand at a birth is a common and good idea, but, where they aren't available, a trained healer or midwife can help almost as much. Furthermore, there's a good chance that healthier women (those with higher Con scores) will have an easier time of delivery than sick women, and it leaves open the possibility that the mother might die in childbirth.