The 21-year-old American will go straight into the Robins' squad for the trip to Newport County on Friday night.
USA youth international Packwood has spent time on loan at Bristol Rovers, for whom he appeared against Cheltenham last season, and Colchester United.
He has been on Birmingham's books since the age of 14, making 18 appearances in the Championship for the Blues.
Packwood becomes boss Russell Milton's fifth signing since taking over at the helm, following Pablo Mills, Matt Sparrow, Shaun Harrad and Danny Haynes.
"I have been looking at the games we have coming up, with three in eight days and I felt we needed cover in every position," Milton said.
"There will be some rotation and Will fits nicely into that. I have had some very good reports about him and I've seen him play a few times myself.
"He is a quick, strong boy, who talks a lot for someone so young. He has played regularly for Birmingham and we now have four good centre halves with Pablo Mills, Troy Brown and Matt Taylor."