Zimbabwe

Cartwright was born in Harare, Zimbabwe. He spent his early life in Marondera, but at the age of 11 emigrated to Australia with his family after their tobacco farm was occupied.

"I don’t remember a huge amount of the bad side of Zimbabwe," said Cartwright. "I remember living on a farm. Our cousins were down the road. It was only in the last year or so when we had to move out of our farm home and live with my mum’s side of the family in Harare was when my perspective started to change."

Move to Perth

His family settled in Perth, Western Australia, where he attended Wesley College.

Cartwright represented Western Australia at the 2008–09 Australian Under-17 Championships, and subsequently progressed to the state's under-23 side in the Futures League, debuting in February 2012.

Cartwright said, "When I was younger, I saw myself as a batting all-rounder, and then I started taking a few wickets and people started saying, ‘’Oh you’re a bowling all-rounder’’ because I was out of form with my batting. So it was quite hard to try to keep the idea of myself as a batting all-rounder in my own mind."

At grade cricket level, he plays for the South Perth Cricket Club. Cartwright also holds British citizenship, and spent the 2010 English season playing for Sidmouth in the Devon Cricket League.

"I used to think 'I’m always going to play for Zimbabwe', but since moving to Australia I’ve actually never thought about playing for the English," said Cartwright.

2012–13

After good form in a Futures League match against the Queensland Academy of Sport, recording a half-century and a five-wicket haul, Cartwright was selected to debut for Western Australia's team for the 2012–13 Ryobi One-Day Cup against Victoria at the Melbourne Cricket Ground, held on 7 November 2012. In the match, Cartwright scored four runs from seven deliveries coming in at number seven in the batting order, and took a single wicket in Victoria's innings bowling medium pace, Western Australia's only wicket of the match.

In December 2012, Cartwright was named in the Perth Scorchers' squad for the 2012–13 Big Bash League season, replacing the injured Pat Cummins. He debuted for the Scorchers in the second match of the competition, against the Melbourne Stars, and top-scored with 17 runs from Perth's innings of 69, the lowest completed score recorded in the tournament. Cartwright made his first-class debut for Western Australia in January 2013, against New South Wales in the Sheffield Shield. He scored 0 and 29 and took 0–19 off two overs.

His second first class game was unusual – he replaced Adam Voges during a game on day 4, when Voges was called up for international duty. Cartwright took 0–22.