I know this is a difficult matter for you, but have you considered just letting the twins grow up without you? If they can be assimilated into the present family and raised in a good home, you'll only be a distraction. I realize they're part of you, but it might be best for everyone if you weren't there as a constant reminder of this woman's infidelity.
It might be best just to cut out and leave the twins in the care of their family. Let them bury the past for the sake of their kids. If you get involved legally, you might destroy the integrity of the family. It makes me ill to think they're going to be raised in that religion, but do you intend to undermine the religious beliefs of the kids while the couple is raising them?
What are your legal rights and wishes regarding this matter?
Since you invited all questions, let me ask a personal one. This woman hid her ring and marital status for a long time. Was she anxious to have a physical relationship? And who initiated it, you or her? Do you think she wanted a long term relationship with you at the time? And why wasn't she on birth control? Many times when twins are involved, women are actually taking fertility meds. Any chance this was the case...that she was trying to get pregnant?
I know you want to be a part of these kids' lives, but like I said, it might seriously damage the relationship of the family. Is it possible you want this woman to leave her husband and marry you, or is that not part of the equation?
Whatever happens, good luck.