It is not a strict requirement for a woman to cover her hair while reciting the bracha for tevilah. However, there is a custom among many women (or mikvah attendants) to place a small towel or cloth over one’s hair for the bracha. In theory, a kallah before her wedding wouldn’t need this covering even according to those who are strict about it, but in practice a mikvah attendant will usually put the towel for the kallah as well.
The basic halacha is that if a woman is not in the presence of men, she does not have to be fully covered for brachot and tefillah. However, there is a halachic concept that one should be dressed appropriately for tefillah, as you are standing in front of the King of Kings (hikon likrat E-lohecha). Some halachic authorities maintain that this includes hair covering for a married woman, other poskim do not require it. However, there can be a practical difference in how one dresses depending on their circumstances – while in bed saying kriat Shma, or while in the mikvah, you do not have to be dressed to the same standards while you are davening under normal circumstances.
We recommend reading R’ Henkin’s book, Understanding Tzniut, as well as www.deracheha.org/head-covering-1-halachic-basis/ for more in-depth discussion of the laws of hair covering.