Aiden Aslin and Shaun Pinner were sentenced by a court which is not internationally recognised, in the so-called Donetsk People's Republic (DPR).
Mr Aslin told his family his captors said there had been no attempt by UK officials to negotiate on his behalf.
The foreign secretary discussed his case with Ukraine earlier in June.
Liz Truss said she and her Ukrainian counterpart Dmytro Kuleba had spoken about "efforts to secure the release of prisoners of war held by Russian proxies" and has called the death sentences a "sham judgement with absolutely no legitimacy".
The Foreign Office is known to be actively investigating cases of British nationals who have been detained in Ukraine, and is providing support to the families of Mr Aslin and Mr Pinner.
Mr Aslin's family said they had spoken to him in a phone call in which he said he had been told "time is running out" by his captors.
"There are no words, just no words, it's got to be everyone's worst nightmare to have a member of your family threatened in this way", Mr Aslin's grandmother Pamela Hall told the BBC.
"Aiden was extremely upset when he called his mother this morning. The bottom line is Aiden has said the DPR has told him nobody from the UK has made contact, and that he will be executed."
"I have to believe what Aiden has said to us, that if the DPR don't get some response then they will execute him. Obviously I hope that isn't true."
Mr Aslin and Mr Pinner moved to Ukraine in 2018, and had served as part of Ukraine's armed forces for several years before the Russian invasion.
Both had Ukrainian partners and had made the country their home. They were captured after spending weeks defending the besieged city of Mariupol.
But the Russian-proxy court called them "mercenaries", accusing them of being sent to fighting in a foreign conflict for money.
They were charged with crimes including the violent seizure of power, and undergoing training to carry out terrorist activities, according to Russian news outlets RIA Novosti.
Earlier this month Ukraine President Volodymyr Zelensky told British students that what Russia was doing to the British pair was a "tragic habit" and said "there cannot be any justifications for such actions".
He said it was "a very powerful and dangerous signal" to other countries that Russia was "acting this way towards citizens of any country".
The UK government has said they should be treated as prisoners of war under the laws laid out in the Geneva Conventions.
"I have cried buckets over this, but crying doesn't help, I want do so something but I don't know what to do.
Ms Hall said: "After Aiden's call this morning what am I supposed to think? I don't want to lose heart but it's very hard."
"I can't help but think that contact should be made between the UK and Russia."
But UK government sources indicate that ministers are currently unwilling to negotiate with Russia directly because it may risk fuelling a false Russian narrative that the men are mercenaries.
Mr Aslin had shortly been due to marry his Ukrainian fiancée, who has come to the UK since his capture.
"He would have been married by now, it's something that I would have dearly loved. I want him to be married and have the life that he wanted," his grandmother said.
"I know that Boris Johnson has been over to Kyiv and spoken again to President Zelensky and I am hoping they spoke about these guys."
In recent weeks Mr Aslin and Mr Pinner have been mocked by presenters on pro-Kremlin television programmes in Russia, which have been showing videos of the men in captivity.
Alex Tobiassen, who also joined the Ukrainian marines with Mr Aslin and Mr Pinner, said the men were now known as "heroes" among many Ukrainian soldiers who continue to fight.
He said their supposed death sentence had galvanised Ukrainian troops even further.
"It actually rallies more people behind them" he said, speaking to the BBC from Ukraine.
"We have family ties here. I'm fighting for my friends, for my family, for my house, and this is absolutely how Aiden saw it.
"Aiden Aslin and Shaun Pinner came here before this invasion started, they made a life for themselves in this country, they settled down.
"They enlisted in the military as a contracted marine, just like anybody who enlists in the military whether in the UK or US.
"These guys went into the office, they went through their medical process, they went through the background checks, they signed up and made an oath for their commitment, these men are not mercenaries, it's clear.
"But, as we see, the Russians just completely have disregard for that."
Regarding the threat of the death sentence being carried out Mr Tobiassen said: "It's absolutely dire, at this point I really hope that the UK government as well as the Ukrainian government is going to step up and figure out some way to negotiate their release.
"That's all I can hope for and that's all that I pray for."
A protest rally is being planned in support of Mr Aslin by the Ukrainian community in Nottingham this weekend.
"There are a lot of people out there supporting Aiden and the other guys who are being held prisoner," his grandmother said.
"I would tell Aiden we all send our love and support, and that there are thousands of people who there who support him and the other guys.
"The people of Newark have been tremendous in their support."