In Colorado (which is an outlier among U.S state court systems), trial court judge in the state court system has a duty to rule on pending motions within three months, and if they do not, a litigant can, after providing certain notices to the judge and appropriate authorities in the court administration office move to have the judge's paycheck suspended until the judge rules.
In practice, even most lawyers don't know that this rule exists. Those lawyers who are aware of this rule virtually never utilize it, out of the well founded belief that it only makes sense to do so if you are a client in a very time dependent situation whom you are certain the judge will rule against when the judge actually gets around to ruling and you want to get on to the appellate process quickly for the client. Even then, the prevailing assumption is that the judge will be prejudiced against litigant in all discretionary matters for the rest of the case, and that the lawyer for that litigant will be at a disadvantage before that judge for the rest of his or her legal career.
Colorado has no similar duty for municipal court judges or administrative law judges (who are not administrative part of the state judicial branch), or for appellate court judges. Likewise, there is no such duty in federal court.
In an extreme circumstance, a state court litigant could file a Colorado Appellate Rule 21 motion (formerly called a writ of mandamus) with the Colorado Supreme Court seeking interlocutory relief from the excessive delay, and a federal court litigant in an Article III federal trial court could seek parallel interlocutory relief from the U.S. Court of Appeals for the Circuit in which the trial court is located. But, either way, the likelihood of the relevant appellate court actually granting interlocutory relief for excessive delay by a trial court in addressing a pending matter is virtually nil, unless the case is one statutorily entitled to priority scheduling such as election law cases, protective order cases, and in Colorado state court, in certain civil cases involving very elderly parties.
By way of example, I have had civil cases in which dispositive motions have been fully briefed and awaiting a ruling from a judge in both a general jurisdiction state trial court, and also in a different federal court, in which no ruling was forthcoming for more than a year after the matter was fully briefed.
It isn't terribly unusual when appellate court rulings are released, to see that the oldest case ruled upon (often an appeal of a serious criminal conviction) that day was filed five or more years before the opinion is issued, even though once a trial court record is transmitted to the court (which typically takes three to six months), and there are typically only three or four appellate briefs to be filed plus half an hour of oral argument to conduct once the case is fully briefed for the appellate judges to consider before ruling.
There are rare cases where proceedings can be even slower due to exigent circumstances.
For example, I once had a limited jurisdiction civil case that went to trial over automobile repairs in which the presiding judge who had heard all of the testimony and taken careful notes from it had a serious health incident the evening after the last day of the trial, leaving him hospitalized for seven months, and in which it was discovered that the tape recorder that was supposed to have recorded the trial court proceedings that could have been used to create a transcript for use by another judge to rule on the case was broken and had recorded nothing. A year after the trial was completely and the judge had recovered and returned to work, we got the judge's ruling that he otherwise would have provided within a week (because that would have been his normal practice, not because he was required to do so) had he not been hospitalized.
Needless to say, no one can tell the U.S. Supreme Court to speed up ruling on a matter in any binding way.
There are some exceptions to this in U.S. criminal law trial practice at both the state and local and at the federal level, where there is a constitutional right to a speedy trial that if abridged results in the dismissal of charges against the defendant, and there are a few other deadlines of a similar character related to the post-conviction process. There is also a very short deadline in criminal cases, once a defendant is arrested, for the defendant to have a first appearance before a judge.