The 23-year-old winger signed for Des Buckingham’s side on loan from Danish team Odense in January, and made five appearances, three of which were starts.
However, he failed to make another appearance after he was taken off at half-time in the 1-1 draw at Blackpool on February 10.
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He now links up with a Cumbrians side sitting inside the League Two relegation places.
Burey made his debut for his new club as a 64th minute substitute in a 1-1 draw at home to Harrogate Town on Saturday.
The Blues, who were relegated last season from League One, appointed former Newcastle United centre back Mike Williamson to replace the sacked Paul Simpson as head coach last month.
Simpson played for the U’s between October 1988 and February 1992, while Williamson spent the 2017/18 campaign at United.
Williamson said on the signing of Burey: “We’re pleased to have Tyler in with us. He’s an exciting player with pace and he feels as though he’s got something to prove.”