Farmer was born and raised in Leytonstone, East London. He has three older siblings; his brother Harry also acts. Farmer took Saturday classes at the Stage One Theatre School from the age of eight. He attended Sylvia Young Theatre School.
Farmer began his career in the music video for Jessie J's "Who's Laughing Now", the play Rest Upon the Wind at the now defunct Tristan Bates Theatre in London and Unity Theatre in Liverpool, and the CBBC series The Ministry of Curious Stuff. He then played Toots in the National Theatre production of Emil and the Detectives.
In 2014, Farmer began starring in the Disney Channel UK series The Evermoor Chronicles (also known as just Evermoor in some countries) as Jake Crossley, a role he would play for both series. Farmer made his feature film debut in 2018 with a minor role in Steven Spielberg's Ready Player One and a voice role in Andy Serkis' Mowgli: Legend of the Jungle for Netflix. In 2020, Farmer appeared alongside his brother Harry in the short film Of Wolves and Lambs and the play For the Sake of the Argument at Bridewell Theatre.

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