在飛機上錄製的一段視頻中，平納先生說，他們是 "靠自己的力量 "逃出來的。
Ten detainees, including the five Britons, were released after Saudi Arabia said it had brokered an exchange between Russia and Ukraine.
A spokesman for the UK-based non-profit organisation, which has been supporting the family of Mr Healy, said they were "looking forward to normality with their families after this horrific ordeal".
In a video recorded on the plane, Mr Pinner said they had escaped "by the skin of our teeth".
"We're now out of the danger zone and on our way home to our families," Mr Aslin said.
Mr Aslin, Mr Harding, Mr Hill and Mr Pinner were captured while fighting with Ukrainian forces and had all been living in Ukraine when Russia invaded in February.
Mr Healy, who had been providing humanitarian assistance in Ukraine, is also known to have been detained by Russian-backed forces at a checkpoint in April, alongside Paul Urey who died in July while in detention.
Their release came as part of one of the most significant prisoner swaps of the war so far, with 215 Ukrainian soldiers released - including commanders of the Azov regiment, which defended Mariupol - by Russia.
In exchange, Ukraine handed over 55 Russian soldiers, as well as the pro-Kremlin Ukrainian politician and oligarch Victor Medvechuk, who was facing charges of treason.
Prime Minister Liz Truss said the news of the release ended "months of uncertainty and suffering for them and their families".
Foreign Secretary James Cleverly welcomed the release of the British nationals but added: "Tragically that was not the case for one of those detained and our thoughts remain with the family of Paul Urey."