Donald Trump has said Boris Johnson asked him to set up a surprise meeting in the White House between the parents of the teenager Harry Dunn and the woman suspected of killing him in a road traffic collision. The president also claimed lawyers may have stymied his efforts to bring them together.

But a spokesman for Tim Dunn and Charlotte Charles blamed Trump’s national security adviser, Richard O’Brien, for the failed stunt and called him a “nincompoop”.

The recriminations broke out after Trump on Tuesday invited the grieving parents to a meeting at the White House and, in what the family described as a bombshell, revealed that the woman involved in the crash, Anne Sacoolas, was waiting in the room next door to meet them and personally apologise. They said Trump pressed them two or three times to go ahead with the “healing meeting”.

Facebook Twitter Pinterest Tim Dunn and Charlotte Charles speak to the media in New York. They visited the White House on Tuesday. Photograph: Eduardo Muñoz/Reuters

The family, who are in the US to raise awareness of their case, refused the invitation, saying they would only meet Sacoolas in the UK where she would also face a police investigation. They said any meeting would need to be prearranged with therapists present.

Sacoolas fled from the UK days after the 27 August accident, claiming diplomatic immunity on the basis that her American husband, Jonathan, worked at RAF Croughton in Northamptonshire. The presence of O’Brien at the White House meeting underlines his background in intelligence.

Referring to the Dunn family not wanting to meet Sacoolas at the White House, Trump said: “She was in the room right out there, we met right here. I offered to bring the person in question in and they weren’t ready for it. I spoke with Boris, he asked me if I’d do that, and I did it. Unfortunately they wanted to meet with her and unfortunately when we had everybody together they decided not to meet. Perhaps they had lawyers involved by that time, I don’t know exactly.”

He described the meeting as beautiful and sad, and said Dunn’s parents were “lovely, desperately sad people”.

On Wednesday a Downing Street spokesperson said: “The prime minister and president spoke last Wednesday. The prime minister asked the president to do all he could to help resolve this tragic issue. The president agreed to work on trying to find a way forward.”

A family spokesman, Radd Seiger, said O’Brien told Charles and Dunn during the meeting that Sacoolas was “never coming back” to the UK. He said O’Brien “appeared to be extremely uptight and aggressive and did not come across at all well in this meeting, which required careful handling and sensitivity”.

Mark Stephens, the family’s legal adviser, said: “Having ice in your veins is a very good characteristic for being a spy but thinking you can treat human beings in that way, you would have to be a nincompoop.

“He’s piled additional grief and hurt on the Dunn family that was entirely unnecessary. He started it by causing her to be a fugitive from justice, spiriting her out of the country on US Air Force transport without telling the Northamptonshire police.” He also said press has been lined up by Trump to see the two families meeting.

Charles said: “We’ve said all along that we are willing to meet her. We are still willing to meet her. But it needs to be on UK soil and with therapists and mediators. And that’s not just for us. That’s for her as well.”

A statement issued by Sacoolas’s lawyer Amy Jeffress said: “We are trying to handle the matter privately and look forward to hearing from the family or their representatives. Anne accepted the invitation to the White House with the hope that the family would meet and was disappointed.”

It is not clear whether Sacoolas is acting freely in refusing to return to the UK or instead is acting on the orders of US intelligence.

Northamptonshire police said they would soon submit a case file on the crash to the Crown Prosecution Service for a charging decision.

If Sacoolas is charged with causing death by careless driving or the more serious charge of causing death by dangerous driving, the force could issue an extradition request or an Interpol red notice. Sacoolas has admitted she was driving on the wrong side of the road when the accident occurred.

The family also has the option of bringing a civil case against Sacoolas. Stephens said: “It is often not understood that diplomatic immunity only applies in the host country, so the UK, not in the home country. It evaporates the moment you set foot on US soil.”