There are two that come to mind right away - Savage Worlds and GURPS. Unfortunately, I don't know how the board game at all, so I can't reference that for mechanics.
Savage Worlds is relatively simple to play and tends to have a cinematic feel to it. There is a specialized cyberpunk book that includes extra traits, information, etc, for that kind of setting, and it's generic enough to probably fit into the NR universe. Don't have it in front of me, or else I would look at it and tell you a bit more.
GURPS is my personal favorite tabletop RPG system, and it stands for Generic Universal Role-Playing System, and it's something you can fit into any kind of game. GURPS is something of a "toolbox", though. While games like Pathfinder, D&D 5th, and Shadowrun are pretty much "open up the book and play", GURPS works best when you can take the time to craft specific things for your setting that aren't included in the book by default. That said, the base mechanic is 3d6 and roll under your skill to succeed, and the fact that it uses a bell curve is extremely pleasing to me. GURPS is kind of front-loaded, as it, it takes a while to actually learn the things that are relevant so you can play, but once you're in the game, it's normally very easy to keep things going, and I find I reference things less than in other games.
Shadowrun deserves an honorable mention for probably being the most well-known cyberpunk RPG system. Now in their 5th edition, it does have the non-netunner inclusion of magic in it's system. However, you could play a game where magic of all kinds is banned, but it might feel like you're ignoring a significant portion of the system. Protip: It's because you're ignoring a significant portion of the system.
Sorry, but that's about the best I can offer. I'm sure there are more cyberpunk settings out there, you just have to look for them.