All pregnancies and all births carry some risk. Obviously that risk varies but the fact that there even is a risk means we should be on a look out as in a medical emergency. An increasing number of births are high risk due to older mothers giving birth and women with health risks and physical or even mental disabilities giving birth
Childbirth is a long and drawn out process that, very often, can take 18-36 hours, or more! This does NOT place it in the general realm of an emergency. Naturally, the mother should be monitored for complications and if it becomes an emergency, they should be moved up to first priority. However, a person with a gunshot wound or a traffic collision victim should certainly be higher up on the charts.
Women have been having babies for hundred's of years. They didn't have doctors, they had midwives, and some only had a neighbor to deliver them. In the 1800's and sooner women would deliver their babies and then head back out to the fields with a baby on their breast, suckling. There was no recovery time. Now we have the luxury of medical doctors, but it still is not an emergency unless the baby is in danger, or if the mother is in danger.