At the hospital i used to work at, it was just the mama that had to consent. Of course once you are out of the hospital either parent could go and get it done without the other parent knowing.
Whats going on mama?
When I had my son, it was only the mother who needed to sign. My husband wanted to circ, and I didn't. I wrote a letter saying I didn't want to and had it put on the chart. I was a little paranoid.
I wonder if your cousin's wife would consider at least waiting a while. It doesn't have to be done in the hospital, so maybe she will wait and think about it some more? She can always have it done later, but she can't go back and undo it, right?