Even adopting a second wife's children is likely to not happen. In many states, mine included, in order for even a biological child of one parent to be adopted by the other parent, a full home study must be done including background checks and they will ensure a legal marriage license in the process. Everyone including all children in the home will be interviewed. If you are living polygamously they will find out and you will be disqualified and the adoption won't happen. AND, just hope they don't remove the children from your home in the process for it. In another state where we used to live, a friend of mine had her legal husband adopt her daughter after the bio dad relinquished rights. They did not have to do a home study but they did have to submit lists of joint assets and a copy of their legal marriage license in order for the judge to place the child.
Short answer, it is a huge Pandora's box and the likelihood of the adoption going through is slim to none. Even if you divorced your first wife and legally married the second wife and then tried, chances are it would be unsuccessful since they would want to interview the "ex" and make sure you are paying (the man) child support and spousal support and see the financial records of such. She could have no evidence of living in the home during the home study and they would ask any children with first wife about their mom or custody or who sleeps where. Ultimately, it is a huge risk and for what, to have you name on a birth certificate of a child? Now a child born to the couple, nobody really glances twice, hospital figures the guy had an affair and got her pregnant and is claiming his child. Adoption though, very sticky and messy in a poly household and it would risk all of the children, not just the ones attempting to be adopted.
Just my two cents. Before poly mindset we wanted me to adopt my step-son and the hoops we would have to jump through ultimately were so many we chose not to do it... and we are legally married and not living polygamously.
Becca