The Right To A Prompt First Appearance
Generally speaking, the deadline imposed by statute is more strict than the U.S. Constitutional requirement.
Many jurisdictions require that a first appearance before a magistrate or judge (also called an arraignment on a complaint) be held within 24 hours of arrest.
Almost all jurisdictions in the U.S. require that there be a first appearance within one or two days that the courts are open after an arrest (usually the day after the arrest unless it is made in the early morning hours, or on Monday if an arrest is made on a weekend, unless the courts are closed on Monday, in which case Tuesday).
See generally, Israel, Kamisar and LaFave, "Criminal Procedure and the Constitution" (1994) at pages 8-10.
The exceptions would typically involve exigent circumstances (e.g. someone arrested at sea or during a hurricane), or a federal government claim that someone is in military detention as an enemy combatant.
The usual and primary legal remedy if someone is detained too long without a first appearance (which is quite rare) is to have a "next friend" bring a habeas corpus action directed at the person holding the individual in custody (typically a sheriff in charge of a jail or a prison warden), while someone is denied a first appearance. This is basically a request for a judge to order a sheriff or prison warden to deliver the prison to a courtroom.
The other remedy (almost always after the fact) is to bring a civil action under Section 1983 alleging a violation of civil rights and claiming money damages against the individuals violating the prisoner's rights to a first appearance and anyone who has instituted an official policy (whether or not in writing) that has the effect of violating a prisoner's rights to a first appearance. The granting of a first appearance does not eliminate the right to money damages in these cases, but usually, the damages award will be very small (apart from attorneys' fees to a prevailing party) except in the most egregious cases. If an official policy is involved, usually this will be done in the form of a class action lawsuit seeking both money damages and an injunction requiring the government to change the policy.
The Right To Be Booked
In addition to the right to a first appearance there is also a statutory right (which probably also has a constitutional dimension) to be "booked" into a regular jail as soon as practicable (or "forthwith") after someone is arrested. In practice, this usually means that someone must be booked within an hour or two of an arrest, absent exigent circumstances (e.g. the patrol car is stuck in a snowbank).
The Right to A Speedy Trial
Once someone has had a first appearance and been charged, a trial must be held within deadlines that are almost always governed by statutes that are more strict than the constitutional requirement. The calculations are rather technical and are often waived with defendant consent.
Criminal defendants who are incarcerated pending trial are entitled to priority relative to criminal defendants who are granted pre-trial release either on bail or on personal recognizance or with a summons to appear.