The foreign secretary, Dominic Raab, has commissioned a review into immunity arrangements for US personnel and their families at an RAF base following the death of Harry Dunn.

He claimed there were “no barriers to justice being done” in the case of the road crash victim, telling MPs the UK government believed diplomatic immunity “clearly ended” for suspect Anne Sacoolas, 42, when she left the country for the US shortly after the crash outside RAF Croughton in Northamptonshire.

Raab added it would be for the Crown Prosecution Service (CPS) and police to decide what steps to take, telling MPs he is “not aware of any obstacle” under the UK/US extradition treaty.

However, Dunn’s family reacted with anger, saying they had no faith in the foreign secretary and his intervention had just added “insult to injury”.

Raab was responding to questions in the House of Commons, including if he had been advised whether the CPS could commence extradition proceedings to return Sacoolas to the UK.

Dunn, 19, died on 27 August after his motorcycle collided with a car being driven along the wrong side of the road near the Northamptonshire base.

Police believe the car was being driven by Sacoolas, a US diplomat’s wife, who they say assured them she would not “leave the country in the near future”. She later left for the US, claiming diplomatic immunity.

The case has sparked an international outcry, with Boris Johnson and Donald Trump getting involved. Dunn’s parents travelled to the US to meet the president and urge Sacoolas to return to engage with the police investigation.

Making a statement to the Commons on the case on Monday, Raab said of the Dunn family: “We will continue to fight for justice for them. I’ve already commissioned a review of the immunity arrangements for US personnel and their families at the Croughton annex holding privileges and immunities under the Vienna convention on diplomatic relations.

“As this case has demonstrated, I do not believe the current arrangements are right and the review will look at how we can make sure that the arrangements at Croughton cannot be used in this way again.”

The shadow foreign secretary, Emily Thornberry, raised several questions, including on why it took so long for the family to be informed that Sacoolas had returned to the US.

She asked Raab to publish all correspondence and records of communications and meetings between the Foreign Office, Northamptonshire police and the Crown Prosecution Service, as well as those with US officials about the handling of the case.

Thornberry asked: “Can the foreign secretary explain why his department asked Northamptonshire police to delay informing Harry Dunn’s family of the departure of Mrs Sacoolas … Why did they not have the right to be told immediately?”

Raab replied: “It was one or two days. The reason we asked for a little bit of time – and this was a request not made by me, I wasn’t aware of it, but by my officials – was to make sure we’d be very clear on what the next course of action would be.”

Earlier this month, it emerged that Raab had sent a letter to the Dunn family in which he suggested that Sacoolas’s return to the US had rendered her diplomatic immunity irrelevant.

Thornberry asked if, given this change, Raab had “been advised as to whether there are any barriers to the CPS commencing extradition proceedings to return Mrs Sacoolas to the UK”.

Raab responded: “Ultimately that must be for the CPS and police to decide and we’re in close contact with them. I am aware there are no barriers to justice being done in this case. And at every stage during this process I have been, and my officials have been, keen to make sure we can remove any obstacles to justice being done.”

He said the government had made its disappointment clear at the US refusal to waive immunity “and we have requested a reversal of that decision at every level of the administration, from the ambassador here through to the representations the prime minister made to the US president”.

Any request for extradition would be made by the CPS, he said, adding: “Under the UK/US extradition treaty, I’m not aware of any obstacle, but I would once again be very mindful of the responsibility I’ve got not to say anything prejudicial.”

The family spokesman, Radd Seiger, told the PA news agency that Dunn’s parents had reacted with fury and anger to Raab’s statement, adding: “It just adds insult to injury.”

Seiger said: “The family just has absolutely no confidence in Dominic Raab. He said we didn’t go to him or his department for help while we were in the United States – that’s because he is the last person we would go to for help after the meeting we had with him.

“The family are angry tonight. They are probably angrier now than they have been at any stage in this entire process.

“Dominic Raab said the ball is in Northamptonshire police’s court for keeping us up to date with the investigation. The chief constable has told us he can’t answer the questions we had for him. Somebody is lying and the family are sick of it.”