WASHINGTON — On Tuesday night, as the House prepared to hold Donald J. Trump’s former chief of staff Mark Meadows in criminal contempt of Congress, a federal judge ruled that the Treasury Department could provide the former president’s tax returns to the House Ways and Means Committee.

It was a sign of progress for Democrats, for sure, but Mr. Meadows could find comfort in the fact that the ruling took nearly two and a half years. And even then, Judge Trevor McFadden of the Federal District Court for the District of Columbia stayed his own judgment for 10 days to give the Trump camp time to file yet another appeal.

The twisting saga of that case — with dozens of motions, hearings, lawyer changes and rulings — gives an indication of how House subpoenas of Mr. Trump’s aides and allies might go as they try to run out the clock on the current Congress and hope for Republican control in 2023, when new House leaders would simply drop the inquiries. The House’s inquiry into the Jan. 6 Capitol attack, a select subcommittee on the coronavirus pandemic and the Ways and Means Committee are all counting on the courts to deliver accountability.

Mr. Trump’s allies — Mr. Meadows, Stephen K. Bannon, Peter Navarro and others — and Mr. Trump himself have perhaps a more realistic expectation: that the slowly turning wheels of justice will deliver nothing of the sort.

“There are people who believe that they can stall and delay themselves out of the truth, and you know, to a certain extent, they have been rewarded,” said Representative Dan Kildee, Democrat of Michigan and a Ways and Means Committee member. “Ultimately, most of us have great faith that justice will be done and the facts will become known. But time is something that works against us, and they know it.”

Mr. Navarro, who coordinated the Trump administration’s pandemic response through his role overseeing the Defense Production Act, said on Wednesday that his refusal to comply with a subpoena from the House Select Subcommittee on the Coronavirus Crisis was dictated by Mr. Trump’s assertion of executive privilege, and thus out of his control.

He said he was not looking to delay a response “until Republicans inevitably take over the House in 2022,” but, he added, referring to Democrats currently in power, “I will beat them on the law, and I will make them look like the fools they are.”

Mr. Meadows, through his lawyer, George J. Terwilliger III, similarly said he was not involved in a cynical game to run out the clock, but had legitimate objections about his constitutional rights and presidential prerogatives. Mr. Terwilliger said that Mr. Meadows had made a “good-faith invocation of executive privilege and testimonial immunity by a former senior executive official,” noting that he had been helpful where he could be, including by turning over thousands of documents the committee had found useful.

But the targets of House subpoenas have been using the courts. Four potential witnesses in the Jan. 6 investigation who were involved in organizing the rally that preceded the violence — Justin Caporale, Maggie Mulvaney, Megan Powers and Tim Unes — filed suit this week against Verizon, trying to prevent the company from turning over cellphone data to the committee.

In their lawsuit, filed in federal court in New Jersey, the witnesses said they were merely “four private citizens” who “voluntarily sat for lengthy interviews and gave thousands of documents to congressional investigators,” only to be rewarded with the panel’s broad demands for “detailed information about their accounts, contacts, personal and political associates, and physical locations.”

Also this week, John Eastman, a lawyer who wrote a memo on how to overturn the election that some in both parties have likened to a blueprint for a coup, sued Verizon and the committee in an attempt to block the release of his phone data, claiming “a highly partisan” invasion of his privacy. Mr. Meadows and Mr. Trump have also sued to block the release of thousands of records, after the former president asserted executive privilege over a vast array of documents.

Several courts thus far have ruled in favor of the committee, but Mr. Trump’s case is expected to make it to the Supreme Court, whose conservative bent is thanks in large part to the three justices the former president nominated.

Some key witnesses have settled on the tactic of invoking their right against self-incrimination. Jeffrey Clark, a Justice Department lawyer who participated in Mr. Trump’s frenzied plans to overturn the election, is scheduled to sit for a deposition on Thursday, and his lawyer said he would invoke the Fifth Amendment in response to certain questions. Mr. Eastman, too, has cited the Fifth Amendment, and a third potential witness, the political operative Roger J. Stone Jr., said he would do so as well, declining to sit for an interview or produce documents to comply with a subpoena.

Representative Jamie Raskin, Democrat of Maryland and a member of the Jan. 6 committee, said a huge difference between the first two years of Democratic control of the House and now is President Biden, who has aligned his White House counsel and his Justice Department with the House’s oversight efforts.

“The legislative and executive branches are completely in agreement with each other, that this material is not privileged and needs to be turned over to Congress,” he said. “So I think things have been moving much more quickly.”

Representative Adam B. Schiff, Democrat of California and another committee member, noted that the House’s two contempt referrals, against Mr. Bannon and Mr. Meadows, were criminal cases. If the Justice Department decides to prosecute Mr. Meadows, as it did Mr. Bannon, both men would face the prospects of jail time and fines.

“And that would be true irrespective of who’s controlling the Congress,” Mr. Schiff said.

With the Meadows contempt referral now at the Justice Department, career prosecutors in the U.S. attorney’s office in Washington will determine whether charges are warranted, and Attorney General Merrick B. Garland will approve or deny their recommendation.

The department moved with relative speed in Mr. Bannon’s case, taking about three and a half weeks to decide that contempt charges were warranted.

But the Meadows case is more complicated, legal experts say, in part because Mr. Meadows had already provided numerous documents to the committee, along with a list of documents that he withheld because of privilege issues. Mr. Meadows was an administration official while he was advising Mr. Trump, and his lawyer has argued that as a former presidential adviser he has immunity and does not need to testify.

The Justice Department has long asserted broad immunity for close presidential advisers, said Jonathan D. Shaub, a law professor at the University of Kentucky who worked at the Justice Department’s Office of Legal Counsel.

Representative Maxine Waters, Democrat of California, suggested that Mr. Navarro could be next.

“If they don’t comply, then we’ve got to get them charged with defying the subpoena request,” she said. “We’ve just got to do it.”

No doubt, the courts are moving more swiftly since the change of power in the White House Counsel’s Office. In two separate rulings — the first in 2019, the second last month — judges said that the Trump White House must cooperate with House oversight demands. But the case two years ago chewed up three and a half months by the time Judge Ketanji Brown Jackson issued a 120-page opinion to end its first stage. Just 23 days elapsed between Mr. Trump’s filing to block the release of Jan. 6 papers and Judge Tanya Chutkan’s ruling against him in November.

But even on a faster timeline, the House’s targets have plenty of judicial recourse.

“The law is not on their side at all, so the only thing they can do is what often happens in litigation, which is to drag it out and seek to delay because the elections are coming,” said William Banks, a professor of law and public policy at Syracuse University and the founding director of the Institute for Security Policy and Law.

And few doubt that a Republican-controlled House would redirect the coronavirus subcommittee away from the Trump administration’s response and toward the culpability of China and Mr. Biden’s handling of the pandemic, drop the quest for Mr. Trump’s tax returns and disband the Jan. 6 committee — or at least vastly shift its focus away from the former president and his allies.

“They want to go after a Mark Meadows because it gets them more airtime in the media than actually helping us understand what went wrong and why the Capitol was so vulnerable that day, and that’s really our job,” said Representative Rodney Davis of Illinois, the top Republican on the committee that oversees House operations.

Some Democrats, seeing the clock ticking, are already showing their frustration. The House has the power of “inherent contempt” — dormant for a century — to compel testimony by ordering its sergeant-at-arms to arrest and jail a refusing witness. Representative Jim McGovern, Democrat of Massachusetts, said recent revelations from the Jan. 6 committee that members of Congress texted Mr. Meadows their ideas for overturning the presidential election and their regrets for failing to do so were “shocking.”

“I’m open to other suggestions here on how to compel people to testify,” he said. On inherent contempt, Mr. McGovern, the chairman of the powerful House Rules Committee, said: “We’ll see. We ought to be thinking about that.”

Katie Benner contributed reporting.