The 19-year-old’s parents, Charlotte Charles and Tim Dunn, had been hopeful of a meeting with Boris Johnson after they said they had “repeated assurances” from senior members of his Cabinet.
Ms Leadsom wrote a letter to the Prime Minister requesting a meeting with the family earlier this month.
Mr Dunn was killed when his motorbike crashed into a car outside RAF Croughton in Northamptonshire on August 27 last year.
Anne Sacoolas, 42, the wife of a US intelligence official, claimed diplomatic immunity after the collision and was able to return to her home country, sparking an international controversy.
The US State Department has said the UK’s extradition request for Mrs Sacoolas is highly inappropriate and would be an abuse.
Speaking about their hopes of a meeting with the Prime Minister, family spokesman Radd Seiger said: “Sadly, it is now clear to the parents of Harry Dunn that, despite repeated assurances to the contrary, Prime Minister Boris Johnson has no wish to, or intention of, meeting with them.
“For all his multiple faults, even US President Donald Trump took the time and trouble to meet with Harry’s parents and extend his condolences personally, albeit for his own personal gain rather than to help Harry’s parents.
“Trump looked the parents in the eye and told them how sorry he was.”
Mr Seiger continued: “Johnson, on the other hand, has done his level best to steer clear of Harry’s parents and continues to do so.
“I have made numerous attempts to arrange the meeting and said we will meet anytime, anywhere and the meeting need not be a long one.
“Each offer has either been rebuffed or ignored.”
Addressing how the parents felt about the Prime Minister, Mr Seiger said: “Unfortunately, the parents’ worst fears about Johnson have come true and they are beside themselves with upset.
“They were deeply suspicious at the outset that he would be more concerned about his relationship with Trump than Harry and them, and that now appears to be the case.
“He had a real opportunity on Tuesday to demonstrate that he has his sleeves rolled up and is right there with his administration doing everything they can to ensure the rule of law is upheld and to bring Anne Sacoolas back.
“He also had an opportunity to make amends and he has chosen not to. Shame on him.”
Mr Seiger added: “The parents will simply move forward now with the dignity and courage they have always shown, determined now more than ever to fulfil their promise to Harry to get him justice.
“They had been looking forward to Johnson giving them his full support in that cause, clearly now in vain.
“Together with their advisers and the support of millions of people around the world, they will continue to fulfil their promise and will never be derailed or go away until their objectives are achieved.
“And unlike the pessimistic, hapless and gaffe-prone Mr Johnson, they know they will achieve them.”
Asked whether Mr Johnson would meet the family, a Downing Street spokesman said: “The legal process is ongoing and the UK has submitted the extradition request.
“The Prime Minister will continue to work to get justice for Harry.”