The 21-year-old joins from Barnet on an emergency loan deal following a recent trial period with the club.
A former West Ham United trainee, Lee enjoyed loan spells at Dagenham & Redbridge and Gillingham before signing for the Bees last summer.
Blues boss Lee Clark told bcfc.com: “We have been scouting Olly for a while now and he was one of Barnet’s standout performers during our Capital One Cup tie earlier this season.
“Olly has impressed us further during a recent trial period and we now look forward to seeing how his development continues over the coming months.”