On 28 March 2013, Uhre made his debut for SønderjyskE. Uhre started on the bench, but replaced Nicolaj Madsen in the 88th minute in a 5-0 away win over Silkeborg IF. In April 2013, Uhre signed his first professional contract, with SønderjyskE.
On 15 August 2014, Uhre was loaned out to Skive IK for the rest of the year. He played 13 league matches and scored three goals. After his loan contract expired in January 2015, he played for SønderjyskE until the summer of 2015. He officially joined Skive IK in the summer 2015. During his sole season under contract in Skive, Uhre scored 15 goals in 33 appearances, only two goals behind topscorer Kjartan Finnbogason.
Return to SønderjyskE
Uhre rejoined SønderjyskE on 5 June 2016, signing a three-year contract with the club. On 14 July, Uhre made his Europa League debut as a substitute in a 2-1 win over Norwegian side Strømsgodset, appearing in the last 27 minutes in place of Troels Kløve. On 17 July, he played his first Superliga match after returning to SønderjyskE, a matchup against AGF. Coming as a 59th-minute substitute, SønderjyskE lost the match 2-1. Uhre made 75 total appearances during his two seasons in SønderjyskE in which he scored 13 goals.
Brøndby
On 18 January 2018, it was announced that a deal had been agreed with SønderjyskE for Uhre to join Brøndby IF on 15 June 2018 on a four-and-a-half year deal. The transfer fee was later revealed to be around 3 million DKK. On 16 July, Uhre made his debut for Brøndby in a Superliga match against Randers FC. The match ended in a 2–0 away win for Brøndby. On 29 July, Uhre came off the bench against Hobro IK substituting Ante Erceg in the 68th minute. Subsequently, he secured a 2-1 away win for Brøndby, scoring the winning goal deep in injury time. He finished the 2018–19 season with 6 goals in 28 league matches. During large parts of the 2018–19 season, Uhre was mostly used as a substitute for regular strikers Kamil Wilczek, Simon Hedlund and Samuel Mráz. On 21 June 2020, he scored the decisive 1-1 equaliser off a bad return ball to seal a draw against FC Copenhagen in the 89th minute. This marked his first goal in 11 months, and came after him coming on as a substitute for Sigurd Rosted in the 84th minute. He would then feature as a starter in Brøndby's following matches, after Hedlund and Mráz had been tested positive for COVID-19. On 9 July, Uhre provided an assist and netted a hat-trick in the club's 4–0 victory over FC Nordsjælland.
