Following successful loan spells with Market Drayton Town and Nantwich Town, Jason Oswell turned professional at Crewe Alexandra in summer 2011. Oswell spent the first half of the 2011–12 season on loan at Rhyl, before he was allowed to move on to Inverness Caledonian Thistle in June 2012. He failed to establish himself in the first-team and left the club in January 2013, ending the 2012–13 season at non-league sides Nantwich Town and Witton Albion. He then spent four months in Australia with APIA Leichhardt Tigers from May 2013, where he won the Waratah Cup, before returning to Wales, signing with Rhyl, though ended the 2013–14 season with Welsh Premier League runners-up Airbus UK Broughton. He joined Newtown for the first time in June 2014, and went on to enjoy a highly successful three seasons with the club, being named as Welsh Premier League Player of the Season in 2016–17 after scoring 22 goals in 31 league games and was twice named in the Welsh Premier League Team of the Year. He signed with Stockport County in June 2017 and was named as the club’s Player of the Season in 2017–18 after scoring 28 goals from 47 appearances. He returned to League football with Morecambe in June 2018.

Oswell then had spells at Wrexham & Stockport County before returning to Newtown in 2023.