The family of a British man condemned to death for fighting Russian forces say they are "devastated".
Foreign Secretary Liz Truss discussed efforts to secure the release of Aiden Aslin and Shaun Pinner with her Ukrainian counterpart on Friday, after the judgment by a Russian proxy court.
Prime Minister Boris Johnson has ordered ministers to do "everything in their power" to secure the release of the two Brits after they were condemned to death for fighting Russian forces.
A statement now issued by the Foreign Office on behalf of the family of Shaun Pinner says: "Firstly, our whole family is devastated and saddened at the outcome of the illegal show trial by the so-called Donetsk People's Republic.
"As a Ukrainian resident for over four years and contracted serving marine in the 36th Brigade, of which he is very proud, Shaun should be accorded all the rights of a prisoner of war according to the Geneva Convention and including full independent legal representation.
"We sincerely hope that all parties will co-operate urgently to ensure the safe release or exchange of Shaun.
"Our family including his son and Ukrainian wife love and miss him so much and our hearts go out to all the families involved in this awful situation.
"We respectfully ask for privacy from the media at this difficult time."
Ukraine's ambassador to the UK, Vadym Prystaiko, suggested negotiations for a possible prisoner swap with Moscow were under way, as it emerged Defence Secretary Ben Wallace made a surprise visit to Kyiv for talks with Ukrainian President Volodymyr Zelensky.
A relative of Mr Aslin urged Britain and Ukraine to "do everything in their power to have them returned to us safely, and soon".
They said Mr Aslin, 28, and Mr Pinner, 48, "are not, and never were, mercenaries" and should be treated as prisoners of war, as they were fighting as part of the Ukrainian army.
The men were convicted of taking action towards violent seizure of power, at a court in the self-proclaimed Donetsk People's Republic.
A No 10 spokesman said: "The Prime Minister was appalled at the sentencing of these men.
"He has been following the case closely and has asked ministers to do everything in their power to try and reunite them with their families as soon as we can.
"We completely condemn the sham sentencing of these men to death. There's no justification at all for this breach of the protection they're entitled to."
Ms Truss said she discussed "efforts to secure the release of prisoners of war held by Russian proxies" during her call with Dmytro Kuleba, Ukraine's minister of foreign affairs.
"The judgment against them is an egregious breach of the Geneva Convention," she added.
Mr Prystaiko believed the two Britons, who he said were targeted for the UK's support of Kyiv's resistance against Russian President Vladimir Putin, will be released in exchange for prisoners held by Ukrainian forces.
The ambassador told BBC News: "It will be a swap.
"The important question is what will be the price for this, because the Russians were talking about some Ukrainian MPs being swapped for them, especially for those who, I now understand, were working for them for all these years."
Mr Wallace discussed how the UK can continue supporting Kyiv "as the conflict enters a different phase" with his Ukrainian counterpart Oleksii Reznikov and Mr Zelensky.
The Ministry of Defence would only say that the two-day visit took place "this week" and it was unclear whether they discussed the men who were sentenced on Thursday.
Russian foreign minister Sergei Lavrov said the convictions were "guided by the laws of the Donetsk People's Republic", the breakaway state controlled by pro-Moscow separatists.
"Because these crimes were committed on the territory of the Donetsk People's Republic, all the rest is speculation," he told a press conference.
The Government was resisting calling in Russia's ambassador to the UK to discuss the case as officials tread a diplomatic tightrope.
There were concerns that making their case an issue between the UK and Russia would assist Moscow in its narrative that the men are "mercenaries" and therefore not entitled to protection under international law.
Britain argues that Mr Aslin, from Newark in Nottinghamshire, and Mr Pinner, from Bedfordshire, are legitimate members of the Ukrainian army and should therefore be treated as prisoners of war.
In a statement to the Newark Advertiser, a member of Mr Aslin's family said: "We love Aiden with all our hearts. He and Shaun, as members of Ukrainian armed forces, should be treated with respect, just like any other prisoners of war.
"They are not, and never were, mercenaries.
"We hope that this sentence will be overturned and beseech the governments of the UK and Ukraine to do everything in their power to have them returned to us safely, and soon.
"We can only imagine what they are going through right now.
"This is a very upsetting development and we ask that our privacy is respected at this time."
A third man, Moroccan national Saaudun Brahim, was convicted alongside Mr Aslin and Mr Pinner.
The men were accused of being "mercenaries" after fighting with Ukrainian troops.
Interfax, a Russian news agency, claimed the men would be able to appeal against their convictions.
Mr Aslin and Mr Pinner were both members of regular Ukrainian military units fighting in Mariupol, the southern port city which was the scene of some of the heaviest fighting since Vladimir Putin's invasion of Ukraine.
Tory former minister Robert Jenrick, who represents the constituency where Mr Aslin lived, called for the Russian ambassador to the UK to be summoned to the Foreign Office.
It comes after a friend of Mr Aslin said the death sentences will "invigorate" those still resisting Russia's advances.
