The 24-year-old seamer, who represents Tasmania in the Australian provincial cricket, has been playing for Lowerhouse in the Lancashire League this summer.
He is expected to feature for Glocesters in the Thursday's NatWest Pro40 fixture against Middlesex.
He will fill the void created by compatriot North, who returned to Australia today after being named in the Australia-A squad for the tour of India.
"I am really excited about the opportunity to play some county cricket," said Drew. "I am looking forward to getting out there today and performing."
Drew has taken 52 wickets at an average of more than 40 in 19 first-class matches, while his 25 limited-overs victims cost almost 29 apiece.