Thinking right now of certain politically crucial cases, but I’m wondering in general - how is it that the courts are so slow, usually, but occasionally, a case is rushed to the front of the line? Who makes those calls? I imagine one of the litigants requests it from the court and then they rule for or against. Can a judge on his/her own, decide a case should be presented “out of order” so as to get more quickly to a ruling? How does all this work?
I was involved in a court case (a voting case) many years ago and due to the time limits specified in the law, the case was heard and decided that Saturday. All the local officials and all the regular court staff had to show up. It was an interesting case. 
Depends on the court, but a party typically can file a motion for expedited consideration by showing that harm could occur if the case were to be adjudicated in the normal time it takes a court to decide a matter. And in federal district courts, at least, judges have a lot of leeway in the management of the docket. Judges handle several cases at once, and the time it takes for an order to be issued will depend not just on when the complaint was first filed, but also on the complexity of the case, the procedural history, and when the judge’s law clerk actually gets around to drafting the order to be signed by the judge.
Don’t the lawyers in the US court systems draft the order? That’s how we do it in Canada.
Maybe in criminal cases. In civil cases (my only area of practice) the only time we draft and submit a proposed order is when stipulated by all parties. At least that’s the general rule—some judges will issue a bench orders and direct the parties to submit and order to be signed.
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Are you talking about setting a hearing for an earlier date, or getting your matter heard before the other cases on calendar that day?
Statutes typically prescribe a short time frame for scheduling trials, but parties can and usually do waive time to give themselves time to prepare. If someone won’t waive time, it has to be done within the statutory timeframes. In California juvenile dependency court, for example, a parent has the right to have their trial within 3 weeks of when their children were detained. But you can’t subpoena witnesses in that time, let alone obtain medical records, hire an expert, determine whether you want to have that expert testify, and get a trial date that works with the expert’s extremely limited availability, so for cases that warrant a full trial, we urge our clients to waive time. There’s a further statutory limit that states in no event shall a trial be continued beyond 6 months, but even that is sometimes ignored when good cause exists and all parties consent. (I have filed a writ petition when the court continued the trial beyond 6 months over our objection, however.)
Different here. The applicant generally filed a draft order as part of the application. If you’re successful on your application in chambers, the judge says " order to go", meaning it can be issued in the same form as the draft order.
In every forum I am aware of, cases can be expedited for various reasons. My understanding is that one party or the other must petition for expedited proceedings. I suspect each court/tribunal’s rules set general criteria under which expedited proceedings are appropriate - risk of imminent/ongoing harm, critical public policy - that sort of thing. It is up to the decision maker to decide whether the criteria are met.
I suspect a party might object to speeding things up, if they feel it would interfere with their ability to prepare.
In my administrative cases, someone can ask that I designate a case as “critical”, which shortcuts various steps, or they can ask that I issue a decision “on the record” - without scheduling a hearing or seeking additional evidence.
I believe in some appeals, the lower court issuing the decision can signal to the appellate court that prompt action may be appropriate.
So basically, someone asks, and the decision-maker agrees.
So that in cases that have direct and potentially immediate effect on the functioning of the government, one of the parties would have to request an expedited hearing? Can’t a judge look at the situation and use that discretion to move something to the head of the line?
No, because in civil matters the parties generally control the pace of the litigation, not the court.
Maybe the parties are in settlement negotiations, and forcing it to trial might prevent them from reaching a settlement.
Maybe one or both is still preparing its case, interviewing witnesses, waiting to receive a report from an expert witness, and so on.
Maybe one of the parties is short of cash and can’t pay for it to go to court yet, and if the court suddenly expedited on its own motion, that party might have to abandon the case.
Maybe one of the parties had to shift its limited litigation resources to some other, more pressing litigation and the other party is content to just let the sleeping dog lie.
Maybe one of the parties is using it to put pressure on the government, and is content to have it hanging over the government to induce the government to change its ways without going to trial.
Judges are not in primary control of the pace of civil litigation and don’t know all the factors behind the scenes that may be contributing to a delay.
Though you might be interested in this
These trials are expedited because they come to trial when the plaintiff is dying.
I’ve had more personal experience in the magistrates court :). As far as I know, hearing dates in our magistrates courts are set entirely by court rules and by the Chief Magistrate. So you could in theory get any kind of expedited hearing, but in practice I’ve never heard of that happening. My first thought would be that if you wanted special treatment of your shoplifting or criminal damage etc charge, you could apply to have it transferred to a higher court.
Well put. To rephrase slightly, as a general rule, judges are supposed to decide a matter based on what the parties present to the court, rather than on their independent knowledge/opinions. That is not to say that a decision-maker is required to ignore his common sense experience or his knowledge of the applicable law. But if neither party suggests to the court that a matter is time sensitive, what basis does the decision-maker have for independently deciding it is?
Also, as a practical matter - there is rarely a shortage of parties who are eager to contend that THEIR case is the most critical matter on the docket and should jump to the front of the line! 
In my practice (quasi-civil), there is no hard-and-fast rule. In the old days, the judge would generally announce rulings orally and direct the prevailing attorney to file a proposed order, after sharing it with the other attorney. Nowadays the judges typically draft their own orders, but some of them still ask for proposed orders to be filed. Occasionally they’ll ask for a party to submit their memorandum of law in Word format (so they can copy and paste from it), which usually means you’re getting a favorable ruling.