Have you ever seen the show Sister Wives? It shows the life of a polygamous man with four wives and like 16 kids, all from the different wives. And from what I saw, all the children accepted their brothers and sisters as true siblings. They grew up this way, so I'm not sure it really has any affect on them, since it's all they've ever known. Taking a child from a monogamous relationship and throwing them into a polygamous one I think would have very harsh consequences. I assume polygamous kids feel their family is the norm since that is how they grew up. They don't see their children as illegitimate, even though we do. I know that isn't the question though..
In bigamy, I would assume it would only become a problem when the spouses found out about each other. I think it would take a lot of psychological therapy to get both the mothers and the children through the issue. I'm sure the children would feel just as betrayed as the wives. It's a very interesting question, and I too look forward to the answers.