From: Rabbi Moshe Sofer
To: Rabbi Shlomo
The Chasam Sofer permits a government-certified Jewish midwife to serve non-Jewish communities both on weekdays and on Shabbos. On weekdays, the permission rests on eivah and the fact that her assistance does not meaningfully contribute to raising a child for idolatry when other midwives are available. On Shabbos, non-melacha attendance is permitted for the same reason, while Torah-level melacha such as cutting the umbilical cord should be delegated to a non-Jew present, unless genuine danger from eivah exists, in which case even Torah-level melacha is permitted.
Source: Shu"t Chasam Sofer YD §131