Buxton's early years were shaped by a mix of cosmopolitan influences. His father, Nigel Buxton, was a renowned travel writer and wine critic who later became a recurring presence on The Adam and Joe Show under the playful moniker "Baaad Dad." His mother, Valerie (n e Birrell), hailed from Chile, adding a multicultural dimension to his upbringing. After spending part of his childhood in Wales, Buxton attended Windlesham House School in West Sussex before moving to Westminster School in London. It was there that he forged lifelong friendships with future collaborators Joe Cornish and documentarian Louis Theroux.
Buxton briefly studied at the University of Warwick but left after two terms to pursue sculpture at Cheltenham College of Art-a decision reflecting his artistic inclinations. These formative experiences laid the groundwork for his unconventional approach to comedy and storytelling.