In 2015, McKenna entered the Glastonbury Festival's Emerging Talent Competition. The festival named him the winner of the contest in April 2015, for which he was awarded a £5,000 prize and a slot on the festival's William's Green Stage. NME called him "one of the most sought-after new acts" in the United Kingdom after his win, shortly after McKenna signed a management contract with Q Prime (the management company that represents British indie rock band Foals, amongst others). More than 40 record companies vied to sign him, with McKenna choosing Columbia Records.
In August 2015, McKenna self-released his first single, "Brazil". It was originally released through his own YouTube channel on December 2, 2014. The song criticised FIFA, the governing body of association football, for awarding the FIFA World Cup to Brazil in 2014 without addressing the extensive and deep poverty affecting the people of the nation. McKenna later told DIY that he wrote the song because "it's politics and what I see in the news, and it's just general things I feel strongly about, things happening in my life." "Brazil" garnered McKenna widespread attention, as many sports commentators found the song to be a commentary on the emerging FIFA corruption scandal. (Ironically, McKenna's single "Isombard" appeared on the soundtrack for the 2016 football video game FIFA 17.) Later in the year, he was interviewed on Sky News to discuss his views on football's relationship with poverty. "Brazil" reached number one on the Alt 18 Countdown for 23 January 2016 on the Alt Nation alternative rock radio station on Sirius XM Radio, and repeated as number one a week later, on 30 January.
In November 2015, following the success of "Brazil" and his shows in London, McKenna self-released his second single, "Paracetamol". The five-and-a-half minute song discussed how transgender teenagers are misrepresented in the media. In an interview with Sound of Boston, McKenna explained that the title of the song came from "the idea of using the lyric paracetamol was a way of comparing the belief that someone can be cured from who they are, via therapy, to an everyday painkiller." Matt Wilkinson of NME called "Paracetamol" McKenna's second break-out hit. Although it was not likely to receive radio airplay or top out the charts, Wilkinson argued, the song showed that McKenna was not just another "UK indie troubadour, a little bit oikish and with a nifty talent for mainstream melody", but rather a solid and accomplished songwriter who can deliver "bruised and vulnerable" vocals. Jon Lyons of the music Web site ThisNewBand.com said the song showed a maturity that clearly indicated McKenna was not "just a teen dream or a one hit wonder." Billboard said in early 2016 that McKenna was "making inroads in America" with "Brazil", which charted on Billboard's Alternative Songs chart (for the week ending 27 February 2016) at number 32, rising to number 26 on the 5 March 2016 chart.