President Volodymyr Zelensky's chief of staff said the bodies of Chris Parry, 28, and Andrew Bagshaw, 47, had now been returned to Ukraine.
No indication has been given as to when they will be handed to British embassy staff to be flown home.
The families of the men said they were killed during a humanitarian rescue.
The two volunteers were last seen heading to the city of Soledar on 6 January.
Mr Bagshaw's family said the pair were attempting to help an elderly woman when their cars were hit by a shell.
Soledar had been the focus of intense fighting and last month Russia's military claimed to have captured the Ukrainian salt-mine town after a long battle. The government in Kyiv disputed the claim.
Despite the continued fighting in Ukraine, Andriy Yermak, Ukraine's chief of staff, said 116 Ukrainian soldiers had been released on Saturday.
He said released prisoners included defenders of Mariupol, partisans from Kherson and snipers from Bakhmut, as well as two personnel from special operations.
Russia's defence ministry said 63 servicemen had been returned as part of a "complex mediation process".
It said it included people of a "sensitive category" thanks to the mediation of the United Arab Emirates, though it did not specify what those people did.
It is the second prisoner swap carried out between the two countries so far this year.
Mr Yermak said Ukraine is continuing efforts to bring everyone home.
Mr Parry and Mr Bagshaw had been in the Donetsk region of eastern Ukraine doing voluntary work.
In a statement issued by the UK Foreign Office last month, Mr Parry's family said his "selfless determination in helping the old, young and disadvantaged" in Ukraine had made them "extremely proud".
"We never imagined we would be saying goodbye to Chris when he had such a full life ahead of him. He was a caring son, fantastic brother, a best friend to so many and a loving partner to Olga," they said.
They added that Mr Parry, originally from Truro in Cornwall, "found himself drawn to Ukraine in March in its darkest hour at the start of the Russian invasion and helped those most in need, saving over 400 lives plus many abandoned animals."
Scientific researcher Mr Bagshaw was a British national but lived in New Zealand. He had been a volunteer in Ukraine since April.
In a statement, Mr Bagshaw's family said last month: "Andrew selflessly took many personal risks and saved many lives; we love him and are very proud indeed of what he did.
"The world needs to be strong and stand with Ukraine, giving them the military support they need now, and help to rebuild their shattered country after the war."
Another foreign national volunteering in Ukraine, Pete Reed, was killed in the eastern Ukrainian city of Bakhmut - the location of heavy fighting in recent days - on Thursday.
Mr Reed, a 33-year-old American volunteer aid worker, was killed "while rendering aid," according to a statement from the humanitarian aid group he founded, Global Response Medicine.
The UK government has previously warned against all travel to Ukraine, saying there is "a real risk to life".
British nationals still in Ukraine should leave immediately if it is safe to do so, it added.