Mehmet was associated with the youth academies at Arsenal, Charlton Athletic, and Manchester United, before he joined the youth academy at West Ham United in 2008. Mehmet departed West Ham in the 2011 close season and had trials with Ipswich Town and Wycombe Wanderers that summer. He then signed for Welling United of the Conference South. Mehmet departed Welling to sign for Turkish Süper Lig club Kayserispor. He departed the Turkish club in January 2015, having played ten cup matches in four seasons. Mehmet then had trials with Luton Town and Leyton Orient in the 2015 close season and then went on to have a trial with Plymouth Argyle in October 2015 and featured for the club in the Under-21 Premier League Cup. He then signed for Scottish Championship club Falkirk in January 2016 after a one-month trial. Mehmet was unable to replace first-choice goalkeeper Danny Rogers and departed the club by mutual consent in January 2017, after informing manager Peter Houston that he wanted to play regular matches elsewhere. Mehmet then signed a short-term contract with EFL League One club Port Vale on 1 March 2017, with the intention of being the back-up to on loan goalkeeper Leo Fasan. Mehmet debuted for Port Vale in a crucial 2–1 win versus relegation rivals Shrewsbury Town at Vale Park on 17 March 2017. One week later, he impressed in a 0–0 home draw versus Milton Keynes Dons, which was the club's first clean sheet in fourteen matches. Mehmet was released by manager Michael Brown, following the club's relegation in May 2017. Mehmet signed a one-year contract with Scottish Championship side Dundee United, after impressing after a trial in July 2017. He was restricted to cup appearances in the first part of the 2017–18 season, despite expecting a better opportunity when Csaba László replaced Ray McKinnon as manager in November 2017, although he explained that the change in management had little effect on him as he spent most of his time at Tannadice with goalkeeping coach Stuart Garden. Mehmet eventually staked his claim as first-team goalkeeper towards the end of that season, as United defeated Dunfermline Athletic in the play-off quarter-finals but had to be substituted for Harry Lewis just under an hour into the second leg, after he injured himself celebrating Scott McDonald's equalising goal. Manager Laszlo stated that "this was the funniest thing he had seen in football and was really stupid". Mehmet signed a new one-year contract with Dundee United in May 2018, however the torn cruciate ligament injury he sustained in the play-offs left him unable to feature during the first half of the 2018–19 season. On 29 March 2019, Mehmet signed for Queen of the South until the end of the 2018–19 season and Queens manager Gary Naysmith explained that he signed him after first-team goalkeeper Martin and third-choice goalkeeper Ryan Gibson both picked up injuries. He rejoined Dundee United on 5 July 2019 signing a two-year contract; sporting director Tony Asghar explained that he was allowed to leave the club in order to prove his fitness, and having done so he was re-signed to provide competition for first-choice goalkeeper Benjamin Siegrist. However Siegrist played all 28 league games in the 2019–20 season as United were declared as league champions with eight games left to play due to the COVID-19 pandemic in Scotland.
