Neil is the narrative director for both games. He basically writes the screenplay, and oversees the mocap and cutscenes. For part 2, he hired Hailey Gross who gets co-writer credit. He gave her full credit for it as they both appeared in the interviews post launch.
Now where it gets iffy is when Bruce Straley gets involved. He was involved in the creation of the IP way back in 2009 after they shipped Uncharted 2 and were told to go and make a new game with a new team. After trying to reboot Jak and Daxter, they settled on TLOU. Bruce has as much to do with this IP as Neil. However, since Neil wrote the screenplay, Bruce was kinda left out even though he had a lot of input on the major story beats.
Hollywood has separate writing credits for situations like this. You get a screenplay credit, AND a separate Story By credit. Bruce would probably get the story credit as a co-writer there. This is basically what Bruce was upset about because when the show went to HBO, Neil claimed sole writing credit of the original when in fact, it was both him and bruce who were told to go create a new game for ND. He was there at the very inception of the franchise.