In 2008, a theater friend suggested to Robinson's parents that Nick should meet with legendary Hollywood talent scout, Matt Casella, who said of Robinson, "He's a real kid, with an extraordinary talent and gift, not to mention a smart, good natured, creative kid with a great smile. Nick is ready for LA." Casella then recommended him to a few agencies and eventually signed with The Savage Agency in Los Angeles. Unfortunately for Robinson, the 2007–2008 Writers Guild of America strike was in full swing so he and his parents returned to Washington where he continued to perform on stage in and around Seattle.
In 2009, after only his second television audition, Robinson landed the role of Ryder Scanlon, nephew of Melissa Joan Hart's character on the ABC Family sitcom, Melissa & Joey, where he acted until the show concluded in 2015.
In 2012, Robinson was cast in the lead role of Joe Toy in director Jordan Vogt-Roberts feature film-debut The Kings of Summer. Filming began in May 2012 in the Cleveland, Ohio-area and ended August 2012. He also guest-starred in the third season of HBO's Boardwalk Empire as Rowland Smith, a young thief who has stolen part of Nucky Thompson's (Steve Buscemi) underground liquor shipments. The episode, entitled “Blue Bell Boy,” aired October 7, 2012. During the summer of 2012, Robinson began starring in a series of television commercials for Cox Communications entitled "Buffer Time is Bonding Time." A new series of commercials began airing in October 2012.
In late 2013, Robinson was cast in Jurassic World, the long awaited reboot of the Jurassic Park franchise which was ultimately released in 2015. He co-starred with Ty Simpkins, playing a pair of brothers whom visit their aunt played by Bryce Dallas Howard at Jurassic World. Jurassic World went on to become the highest grossing film of 2015.
He was also cast as Ben Parish in the film adaptation of the novel The 5th Wave which is due for release in January 2016. Robinson's roles in both Jurassic World and The 5th Wave resulted in his character being written out of Melissa & Joey for multiple episodes of the show's final season. With sporadic appearances throughout, he returned for the final three episodes. He was also cast as the lead in upcoming independent film Being Charlie which premieres at the 2015 Toronto International Film Festival.