Podmore was born at Hammersmith and educated at Twyford High School. He began playing cricket at Ealing Cricket Club and played for Middlesex from under-15 level onwards, as part of the Middlesex Academy in 2011 and 2012. He was part of the MCC Young Cricketers programme at Lord's and signed his first professional contract with Middlesex in September 2013.
Senior cricket career
Podmore made his Middlesex debut as a 19-year old in the 2014 NatWest t20 Blast against Essex, after a poor run of results for the team led to changes in the side. Podmore took his opportunity as Middlesex's lead bowlers were kept out of the T20 side during the summer and went on to play in the 2014 Royal London One-Day Cup 50 over competition later in the season. After taking 13 wickets in the same number of matches he extended his contract with Middlesex at the end of the season. He played in only five T20 matches the following season for the First XI, but took 42 wickets for the Middlesex Second XI and his contract was once again extended at the end of the season. At the start of the 2016 season, Podmore joined Glamorgan on loan for the first six games of the 2016 County Championship season. He played in four County Championship matches for the Welsh county, making his first-class debut against Kent at the St Lawrence Ground in Canterburyat the beginning of May. After returning to Middlesex he made four County Championship appearances for the county as they won the Championship. In April 2017 he returned to Glamorgan on loan, playing twice for the county before returning to Middlesex. After playing four one-day matches for Middlesex, he briefly joined Durham on loan during August 2017 to play for their Second XI with the possibility of a permanent move, before joining Derbyshire on a 28-day loan later the same month. After the 2017 season Podmore had surgery to release a trapped nerve in his bowling elbow and spent time in South Africa playing for Union Cricket Club in Port Elizabeth. In April 2018 Podmore left Middlesex to join Kent on what was reported to be a three-year contract. Podmore had an agreement with Middlesex that he could move to another county if a suitable contract became available. Podmore's first season with Kent was branded as "successful". He played in each of Kent's County Championship matches, taking 43 County Championship wickets at an average of 23.30 runs per wicket. He was the county's second leading wicket-taker in the Championship as the side won promotion from Division Two and took his best first-class figures of 6/36 against Middlesex, his first five-wicket haul. At the end of the season he signed a contract extension, keeping him under contract at Kent until the end of the 2021 season.