Wife of Kohen giving birth:
A. Wife of Kohen giving birth in hospital:
There is no problem with the wife of a Kohen giving birth in a hospital. Furthermore, she may remain in the hospital as long as necessary and there is no need to take her out as soon as possible.
B. Husband who is a Kohen accompanying his wife in hospital:
It is permitted for the husband of a woman to accompany her to the hospital even if he is a Kohen, if his wife desires him to be with her for the sake of helping her and calming her during her birth. Nonetheless, he should not roam around the hospital unnecessarily. [If one’s wife needs her husband to be by her side in order to calm her for the birth, then he may be by her side even if he knows for certain that there is a Jewish corpse in the vicinity. However, in such a case he must leave the hospital as soon as his wife no longer needs him.]____________________________________________________
 See Toras Hyaoledes 7:8-12
 Betzel Hachochma 3:102; Shevet Halevi 6:175; Encyclopedia Hilchatit Refuit p. 28-30; Nitei Gavriel 62:12; Toras Hayoeldes 60:1-3; See Even Yisrael 8:77 that deals with this question in regards to giving birth in a hospital that is not careful about Tumas Meis. He concludes there that ultrasounds are not 100% accurate, and that it is for the needs of a Yoledes who is in Sakana and hence she may choose to go to whatever hospital she wishes. Nevertheless, she is not initially to take an ultrasound and hence remove the Sfek Sfeka. This is all with regards to which hospital to give birth and does not relate to a pregnant woman entering a Beis Hakevaros if she knows the child is male. It is understood that in such a case there are more Poskim who rule stringently.
 The reason: As by a Kohen child who is below the age of understanding there is no obligation to separate him from impurity. [Admur 343:3 “So too if he is a Kohen, there is no obligation to remove him from a home that contains impurity unless he has reached the age of understanding if which case his father is commanded to take him out.”; M”A 343:2; M”B 343:3] However, see Toras Hayoledes ibid footnote 13
 Nitei Gavriel 56:23;
 See Toras Hayoledes 7:9 footnote 14