The US diplomat's wife granted immunity after the crash which killed teenager Harry Dunn is "devastated by the tragic accident", her lawyer has said.
Anne Sacoolas's legal representative, Amy Jeffress, said she would "continue to co-operate with the investigation".
Mrs Sacoolas, 42, left for the US under diplomatic immunity despite being a suspect in the crash with Mr Dunn, 19, in Northamptonshire on 27 August.
But the Foreign Office said, having gone home, she no longer has immunity.
A statement issued on behalf of Mrs Sacoolas, whose husband worked at RAF Croughton in Northamptonshire, said: "Anne is devastated by this tragic accident.
"No loss compares to the death of a child and Anne extends her deepest sympathy to Harry Dunn's family."
It added she had "fully co-operated with the police".
"She spoke with authorities at the scene of the accident and met with the Northampton police at her home the following day. She will continue to co-operate with the investigation," the statement continued.
"Anne would like to meet with Mr Dunn's parents so that she can express her deepest sympathies and apologies for this tragic accident.
"We have been in contact with the family's attorneys and look forward to hearing from them."
Foreign Secretary Dominic Raab wrote to Mr Dunn's family on Saturday to explain both the British and US governments now considered Mrs Sacoolas' immunity irrelevant.
The letter said: "We have pressed strongly for a waiver of immunity, so that justice can be done... Whilst the US government has steadfastly declined to give that waiver, that is not the end of the matter.
"We have looked at this very carefully... the UK government's position is that immunity, and therefore any question of waiver, is no longer relevant in Mrs Sacoolas's case, because she has returned home.
"The US have now informed us that they too consider that immunity is no longer pertinent."
In response to the letter, Mr Dunn's mother, Charlotte Charles, said: "We've known from the start that the "extra" feeling in the pit of our tummies, told us that something wasn't right.
"We're proud of ourselves for fighting for justice for Harry, and not ignoring this gnawing within our bodies.
"We'd rather have our beautiful boy back, but we are also elated that all this fighting for justice for Harry has not been in vain.
"We'll continue to fight for change to the diplomatic immunity laws and any other positive changes we can achieve."
Mark Stephens, a lawyer for the Dunn family said: "She was allowed to, or encouraged to be spirited away on an American transport plane and effectively rendered a fugitive from British justice.
"And now of course we find out that she's not entitled to diplomatic immunity, and in those circumstances she is in a foreign land a fugitive from British justice.
"We do hope she returns herself voluntarily and that this was just a bad piece of advice she received from the American authorities."
What is diplomatic immunity?
About 23,000 individuals in the UK have diplomatic immunity, a status reserved for foreign diplomats and their families, as long as they don't have British citizenship.
It means that, in theory, they cannot face court proceedings for any crime or civil case.
However, where crimes are committed, the Foreign Office can ask a foreign government to waive immunity.
Diplomatic immunity is by no means restricted to those named on the Diplomatic List. Drivers, cooks and other support staff who have been accredited to Britain ("the receiving state") have the same diplomatic status and immunity.
Earlier, the lawyer for Mr Dunn's family, Radd Seiger, appealed for anyone with information about Mrs Sacoolas's return to the United States to come forward.
Mr Dunn's parents, who have previously said they are considering civil action against Mrs Sacoolas, are set to fly out to the US on Sunday and will visit both New York and Washington DC.
Mr Seiger said they would be "engaging with the media and politicians as they reach out for support from all Americans and to ask them to put pressure on the US administration to do the right thing".
On Friday, Prime Minister Boris Johnson said the US was "absolutely ruthless" in its safeguarding of Mrs Sacoolas following the decision to grant her diplomatic immunity.
He said although President Donald Trump was sympathetic towards Mr Dunn's family's views on the use of diplomatic immunity, the US was "very reluctant" to allow citizens to be tried abroad.
That followed the revelation that Mrs Sacoolas would not return to the UK when briefing notes held by Mr Trump were photographed at a White House news conference.
Chief Constable of Northamptonshire Police Nick Adderley has said the investigation into the crash will continue.
The force has said CCTV of the crash in which Mr Dunn died shows a Volvo travelling on the wrong side of the road.