Scheduling and workload
Courts the world over are underfunded which means that judges and their administrative staff are overworked. Like all of us who are overworked, they prioritize their work which means that urgent stuff gets done before non-urgent stuff.
Broadly speaking, courts have two roles: stopping bad stuff happening in the future (warrants, injunctions etc.) and dealing with bad stuff that's already happened (sentencing, civil judgments etc.). Typically preventing bad stuff from happening is more urgent than dealing with bad stuff that's already happened.
Therefore applications for warrants, injunctions etc. are typically prioritized and turned around quickly - typically the facts and legal issues are not overly contested or complex and so they can be dealt with speedily.
Making and writing judgments (which in my personal experience are the same thing - in writing the decision you make the decision and the decision can turn out being different from what you thought it would be when you started) are time consuming and, from the point of view of the justice system, not urgent. In a criminal case, it doesn't really matter if the convicted criminal starts serving their sentence next week or next year. Similarly in a civil claim, the damage has been done and interest will be paid to the winner by the loser to compensate for the delay in receiving it.
This is cold comfort for the litigants who are left waiting but there are only 24 hours in a day summer and winter alike and even judges need to sleep.
As for research, this is not really the judge's role in a common law court. Its for the prosecution/plaintiff and defendant to direct the judge's attention to the relevant law and to provide submissions on why they believe their interpretation is correct and the other side's is nonsense. A common law judge at first instance is there to decide the particular dispute, if they have a question in their mind about the law they will generally call for the parties to make submissions on it. That said, judges's are generally experienced lawyers who work in their field day in and day out - just like other professionals (e.g. doctors, engineers etc.) they know 95%+ of their subject matter backwards and, while they will refer to the legislation and cases in order to quote it, they rarely have to do research from first principles.
the parties in a case don't always do what they committed to do by the time that they committed to do it (shocking I know). There can be legitimate and illegitimate reasons for this. So parties will ask for delays and continuances.
A judge has to decide whether to grant these requests, balancing the competing tensions of efficiency and procedural fairness all within the overarching scheme of "justice" for everyone involved. Of course, a judge wouldn't be human if seductive little thoughts like "If I grant this continuance, I'll be able to finish writing that other judgement" didn't pop into their head.