In the United States, there are two main sets of laws governing the right to a speedy trial.
The right to a speedy trial is protected by the Sixth Amendment, but that right is somewhat amorphous. It does not entitle you to a trial within any specified period of time, only to a trial within a reasonable amount of time, considering four factors, "[l]ength of delay, the reason for the delay, the defendant's assertion of his right, and prejudice to the defendant." Barker v. Wingo, 407 U.S. 514, 530 (1972).
The right to a speedy trial is also statutorily protected. As far as I know, every state has a law that sets bright-line standards for how long the government has to bring a case to trial. For instance, the federal Speedy Trial Act generally entitles a defendant to a trial within 30 days of arrest. 18 U.S. Code § 3161(b).
But the statutes also generally allow quite a bit of flexibility. The defendant himself will often want to delay the trial while he pursues a fact investigation, discovery, pretrial motions, etc. But the government may also be entitled to delay the trial, especially if it is unable to prepare for trial because it is busy responding to the defendant's discovery requests or pretrial motions.
With both the constitutional and the statutory right to a speedy trial, then, courts are generally permitted a bit of latitude to delay a trial as necessary, especially in cases where the delay is not attributable to the government.
With COVID-19, the widespread practice has been, as you noted, to delay trials. Like any other, that right is not absolute, and it must give way where the force of nature. Although inconvenient for many defendants, there was little suggestion -- at least in the early days of the shutdown -- that this practice violates the right to a speedy trial in general, though there may be cases where specific facts lead to a different outcome. But as time wears on, it seems likely that prosecutors and judges will need to find a way to make trials happen.