Generally, It can be done any time a case is pending. But please remember that there are lots of strategic reasons NOT to ask for a speedy trial. If your husband is held, the charge must be very serious. You should address this with your husband's lawyer, and respect his or her judgment on this issue. Good luck.
The response I have provided is general in nature, and does not create an attorney-client relationship. My practice is based in Rhode Island, and the law and practice in other states or jurisdictions may be different.
A statutory demand must be filed after the return of the indictment. If he is not taken to trial in time, after properly filing it, the case is dismissed. As to his constitutional demand, he can file the demand at any time. It actually is a bit convoluted how the two rules differ. Before you go demanding a speedy trial, you had better be sure that you want a trial to happen really fast because that is what is going to happen. I agree that one best speak with his attorney.
The information given is not intended to be legal advice and is based upon limited facts. No attorney client relationship is created and you should not treat the comments given as legal advice.
Bulloch County has 4 "terms of court", each lasting 3 months. You are currently in the August term that lasts through the beginning of November. If he has not been indicted or formally accused then he would have to file a speedy trial demand under the federal constitution. If he was indicted more than two terms ago (three if a murder case) then he would have to file under the federal constitution. If within those two terms (or three if murder) he could file a demand under a Georgia law that would require them to bring him to trial in, well, a range of months or terms that his attorney could probably describe to you.
One trick we use to get things started is demanding action once he has been in jail for more than 45 days without an indictment. There's no automatic legal remedy but it's great to go into court to put the heat on the prosecutor to bring the case. Then you get deals! (except with murder, they get all picky with that).