Oelschlägel began his career in 2001 at Dynamo Dresden. From 2006 to 2008 he played for SG Dresden Striesen, before returning to Dynamo. In 2012, he joined the youth system of Werder Bremen. In the 2014–15 season Oelschlägel moved to the second team, who at that time played in the Regionalliga Nord making his debut on 16 November 2014 debut in a 4–0 away win against FT Braunschweig.[2] In 2015, he celebrated promotion to the 3. Liga with the second team.[3]
After Oelschlägel went to the first team's training camp in Neuruppin in July 2015,[4] he signed a professional contract with Bremen in January 2016 which lasts until 30 June 2018.[5] With multiple goalkeepers injured, Oelschlägel was Felix Wiedwald's backup in the first team's match against Schalke on 24 January 2016.[6] Having suffered an injury early in the first half of the 2015–16 season[7] he played most second-team matches in the second half of the season.[8]
On 20 June 2018, Oelschlägel signed for Borussia Dortmund on a one-year deal from Werder Bremen for an undisclosed fee as third choice keeper, playing for Borussia Dortmund second team in the Regionalliga West. In February 2019, he played his first match with the first-team, appearing against his former club Werder Bremen in the DFB-Pokal with first and second goalkeepers Roman Bürki and Marwin Hitz not fit to play. Oelschlägel made multiple saves during the match before making a mistake which resulted in Werder Bremen equalising for 3–3 in extra-time. Werder Bremen went through on penalties with Oelschlägel unable to save a penalty.[9]
On 5 October 2020, Oelschlägel moved to FC Utrecht on a free transfer, having agreed a one-year contract.[10]
On 15 July 2022, Oelschlägel joined FC Emmen on a two-year deal.[11]