The good news is that most facilities do not restrict visitation strictly to immediate family. The definition of who can visit is broader than people often assume.
As long as you pass the background check and are not a convicted felon or former inmate, there is a strong chance you can get approved as a visitor. Facilities run every applicant through a criminal database check as part of the standard approval process, and a clean record is the primary requirement. The relationship to the inmate matters less than most people think.
The fact that you are actively involved in his life works in your favor. Facilities generally want inmates to maintain positive outside connections, and an engaged family member who is clearly invested in someone's wellbeing is exactly the kind of visitor the system is designed to accommodate.
The right first step is to call the facility and ask to speak with his counselor. Get the specific visitation rules for that institution, find out what the application process looks like, and ask directly whether extended family members are permitted. Rules vary enough between facilities that getting it straight from the source saves you time and avoids showing up unprepared.
Once you have that information, complete the visitor application and submit it. The approval process takes a few weeks in most cases, so the sooner you get started the sooner you are sitting across from your nephew.