There may be two questions sort of embedded in here: (1) How long does the government have before it violates Speedy Trial; and (2) How long does the government have to turn over its evidence? Unfortunately, they're probably both impossible to answer at this point.
Under the Speedy Trial Act (18 USC 3161):
In any case in which a plea of not guilty is entered, the trial of a defendant charged in an information or indictment with the commission of an offense shall commence within seventy days from the filing date (and making public) of the information or indictment, or from the date the defendant has appeared before a judicial officer of the court in which such charge is pending, whichever date last occurs. If a defendant consents in writing to be tried before a magistrate judge on a complaint, the trial shall commence within seventy days from the date of such consent.
IRA was indicted on February 16, 2018, so 70 days later would be April 27, 2018, which has obviously already come and gone. But their initial appearance wasn't until May 9, 2018, which pushes the time back to July 18, 2018.
That would be the default latest date for a trial to begin, but the Act also allows delays for a variety of factors. Among them are delays "resulting from other proceedings concerning the defendant," which includes time the court spends considering the defendants' motions. So if IRA moves to dismiss the case, all the time spent fighting over that motion would "toll" speedy-trial time, meaning that it would stop the clock until the motion was decided.
Another provision of potential relevance is 18 USC 3161(h)(8):
Any period of delay, not to exceed one year ... [if] an official request .. has been made for evidence of any such offense and that it reasonably appears, or reasonably appeared at the time the request was made, that such evidence is, or was, in such foreign country.
So if there's evidence somewhere in Russia linking these guys to the crimes they're accused of, that might push speedy trial back by a year just to deal with that issue alone.
So the short answer to Question 1 is that Speedy Trial requires the trial to begin on July 18, but that practically speaking, that probably won't happen.
The timing of discovery, the process of each side turning over its evidence to the other side, varies based on the kind of evidence involved, so there are usually multiple disclosures of evidence between the time of indictment and the actual trial.
Under current DOJ guidelines, prosecutors are encouraged to provide "broad and early discovery." More specifically:
Exculpatory evidence available under Brady v. Maryland should be turned over "reasonably promptly";
Impeachment evidence available under Giglio v. United States should be turned over "at a reasonable time before trial to allow the trial to proceed efficiently"; and
Material available under Rule 16 should be disclosed "as soon as is reasonably practical."
As you can see, these aren't really hard deadlines. The defense will ask for the documents, the government will turn over documents, and then they'll probably both fight about turning over some other documents, and then they'll wait for the court to decide what to do.
So when does Mueller have to turn over his evidence? Probably the best answer we have right now is: "When a judge tells him to."