A woman should recite her own beracha when the baby is born, though she is permitted to choose to have her husband say it for both of them. Technically, a woman may recite a beracha when undressed, so long as her genitalia are not exposed. (See more here.) This is easily enough accomplished by throwing on a blanket, so should not cause a delay. If she can, she should also try to make sure that her hands are clean, though washing them is not obligatory.
The issue of the berachot is of particular interest because the Talmud tells us that we recite “ha-tov ve-hameitiv” upon birth of a boy because that parents have shared benefits from a boy’s birth (Berachot 59b). The implication of not mentioning daughters in this context is that these outweigh those from a daughter, a classic example being how a boy grows to help his parents in old age (Shut Ha-Rashba 4:77). This idea may be tied to the Halacha that a married daughter is released from Kibbud Av Va-em, whereas a married son is not (Kiddushin 30b). In any case, some modern halachic authorities, including Rav Nachum Rabinovitch and Rav Yosef Tzvi Rimon* have argued that nowadays, parents who feel true shared benefit upon birth of a daughter may recite ha-tove ve-hameitiv. Others argue that even today we should follow Mishna Berura and recite She-hechiyyanu upon a daughter’s birth, because the joy is on the whole more personal.
We hope to address the berachot recited after childbirth in greater depth in the future.
*We thank Rav Yonatan Ziring for bringing these two sources to our attention.