(CNN) Where are the children? And what's taking so long?

Devastated parents are still searching for their kids . Officials are pointing fingers over who's responsible and have yet to release details about how families will be reunited.

Immigration attorneys and rights groups say that's because officials still don't have a plan to solve a crisis the government created.

"All we get is bureaucratic doublespeak, indifference and excuse-making," said Frank Sharry, executive director of the immigration advocacy group America's Voice.

Deadlines loom

As of Monday, 2,047 immigrant children who had been separated from their parents remained in government custody, according to the latest figures released by the Department of Health & Human Services.

An agency spokesman declined to provide an updated figure on Thursday, saying officials would only provide the total number of immigrant children in custody. That figure, 11,869, includes both children who crossed the border alone and kids who were separated from their parents.

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On Tuesday, HHS officials told reporters they were working on reunifying children and parents as soon as practicable.

"We have always known where the children are," said Commander Jonathan White, assistant secretary of preparedness and response. Officials are working on facilitating communication between parents and kids, he said, and linking records between different government systems.

The clock is ticking. A judge's ruling Tuesday laid out a series of deadlines the government must meet:

• Within 10 days (by July 6), officials must make sure every separated parent has a way to contact their child.

• Within 14 days (by July 10), children under 5 must be reunited with their parents

• Within 30 days (by July 26), all children must be reunited with their parents.

"The United States government has more than enough resources to get this job done as long as it treats it as an urgent priority," said Lee Gelernt, deputy director of the ACLU's immigrants' rights project.

And in most cases, Gelernt said, the reunifications should be able to happen before the deadline.

"Up until now, the reason the reunifications haven't occurred swiftly is because there was no plan or intention to do so," he said. "We'll see how long it takes now that the court has ordered them to do it."

Gelernt, the lead attorney in the ACLU's lawsuit over family separations , says the reason it's urgent is clear:

"There are little children who are being traumatized every day they're separated from their parents, crying themselves to sleep, wondering whether they're ever going to see their parents again," he said.

Moving the goalposts

Several parents in immigration custody told CNN this week that they're frantically searching for their children and unsure of where to turn.

Speaking in a phone interview from the Cibola County Correctional Center in New Mexico, Diego Pascual Andres said Thursday that he has no idea where his 13-year-old son is.

"I'm very sad. ... I want to know where he is and if he's doing all right. I'm worried about him," Pascual said before the phone call cut off.

Across the country, advocacy organizations and lawyers are sharing stories of other parents who are desperately trying to find, reach and reunite with their children.

"It's a nightmare scenario," said Michelle Lapointe, acting deputy legal director for the Southern Poverty Law Center.

The organization is working to help about 20 men held at the Stewart Detention Center in southern Georgia reunite with their children.

"Some have had no communication and don't even know where their children are," she said.

Others have phone numbers of social workers and have been able to make some contact. But each call gets met with a different response, Lapointe said.

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"There's just no system. There's no one standard way of getting this information about their children to men who remain in ICE custody," she said. "It's incredibly difficult and it's incredibly demoralizing for them, and I can't even imagine for the children."

Even parents who've tracked down their kids are having trouble getting officials to release them, some attorneys allege, claiming the process isn't clear and rules keep changing.

"It's a seemingly constantly moving goalpost," attorney Britt Miller told reporters in Chicago on Thursday morning, standing beside a client who sued the government to reunite with her 9-year-old son. "When she first came into the country, it was you're going to be released, and here's the 800-number to find your child. The 800-number got her nowhere. She independently found out and discovered where her child was, because the 800-number that they gave her, no one answered."

'I cried almost every day'

On Thursday afternoon, a federal judge ordered the boy's release from custody. And by Thursday evening, mother and son spoke about their experience as TV cameras rolled.

Lidia Souza, who said she fled Brazil and is seeking asylum, told reporters it was terrible to be separated from her son, Diogo. The boy spent a month in government custody at a Chicago shelter.

"I cried almost every day when I wasn't with my mother," Diogo said.

He described eating meals at the facility, playing with other children in a classroom and spending some time quarantined when he had chicken pox.

A roomful of reporters peppered him with questions.

What was it like spending your birthday there?

"Very sad," he said.

And what about the other children?

"The other children," he said, "are suffering a lot."