DaasYochid -"However (and I may surprise some with this opinion), I think that even if she never intended to cover her hair, her geirus is still valid. We don't need to find a legitimate heter to leave the hair uncovered, all we need to do is be able to assume that she doesn't think it's a violation of halacha. That's reasonable to assume, since so many shomrei Shabbos ladies don't wear a head covering.
Tosofos in Shabbos (68A bd"h Ger - http://hebrewbooks.org/shas.aspx?mesechta=2&daf=68&format=pdf) say that even if someone doesn't know about Shabbos, it doesn't passul the geirus.
I was present when this very shaila was presented to an adam gadol; there was a lady who was m'gayer with no intention to cover her hair (she converted with her husband). The godol affirmed the geirus."
DyafMaven brought up this point already. I quickly said that there is a difference between the two cases. But now I'll elaborate. In the case of Shabbos there was no way for him/her to know about Shabbos because she/he lived amongst Goyim. Here in this case she lives amongst Jews, but the Jews don't follow Halacha. So is this called being Mekabel all the Mitzvos or not? It could be that the case with the Odom Godol was where she didn't have knowledge about covering her hair due to where she lived. In this case the lack of knowledge is based on s/o telling or informing her or her observation of MO Jews that there is no Chiyuv to cover her hair. Maybe wrong knowledge is worse than No knowledge at all and her Geyrus isn't good? Ask the Shaila to this Godol, if you won't post his name so that I could ask him.