Contact the DFCS caseworker. If the child was removed from his mother, there is already a case in the Juvenile Court of the County where the mother lives. The DFCS caseworker is in charge of making sure that the child is in a safe place and a good home.
Talk to the caseworker very straightforwardly, calmly and un-emotionally, Let him/her know anything that you have in your background that might make you appear less-than-perfect (like arrests of prior DFCS case yourself...). Be prepared to show a safe and healthy home, and understand that if your fiancee is in the home, that the same rules apply to him (her?), too. If you have kids, hopefully they reflect well on your parenting skills.
But the fact that the kids are in DFCS custody gives them and the juvenile court all the power. Understand that, and be prepared for a "no" answer. But I suggest that you try- that way you have no personal regrets for not trying. I've been there myself.
Also, understand that getting your nephew can sometimes drive a wedge between you and his mother- and if there's already a wedge there, it can get bigger. DFCS is aware of that, and will consider such subjective issues in making a decision. Their goal and the Court's goal is the "best interest of the child", so understand that, ultimately, the decision is not personal.
You can contact me at my office at 770-744-4211 or by email through this website. The questioner and any reader do not have an attorney-client relationship formed by our communications on this website. Advice given by me on this website is general advice based on partial information. You should not rely on any advice given without first hiring a lawyer in the area where the case is pending, and providing that lawyer with full information.
There are no forms that you will need prior to contacting DFCS. Their presence in his life indicates that there is already an open deprivation case so there wouldn't be anything for you to file. You just need to be certified by DFCS as an appropriate temporary placement/custodian for your nephew.
Your mother should also notify your nephew's case worker about the food stamps issue. He/she may be able to assist her with acquiring benefits for him.
The above answer is a general explanation of legal rights and procedures. It does not constitute legal advice. Nor does it establish an attorney-client relationship between the individual posting the question and the attorney providing the answer.