Soohail Shaikh said his cousin "hasn't given up hope".
Despite that "his feelings and his conversation with us included some items which we strongly feel were still not rational and further go on to stress in our minds that he does need some medical assistance," he said.
Mr Shaikh has been detained in hospital since August, suffering from high blood pressure.
Meanwhile, supporters are gathering outside the Chinese embassy in London. They will deliver a letter to the ambassador calling for a stay of execution.
It has been organised by a group formed on the social networking site Facebook called Stop The Execution of Akmal Shaikh.
Maya Farr, 18, who helped organise the vigil, said she was "opposed to the death penalty".
"But there are so many aspects to this case which we are really opposed to," she said.
"His mental history has not been assessed."
Soohail Shaikh after visiting his cousin Akmal Shaikh in prison
Legal charity Reprieve has been working with Mr Shaikh and his family.
The organisation said the Foreign Office had done a "huge amount" for Mr Shaikh and it had made 10 approaches to the Chinese government in the past six months.
Sally Rowen, legal director of Reprieve's death penalty team, told the BBC Home Secretary Alan Johnson had said on Monday that "efforts would be intensified".
The Chinese authorities had assisted the cousins' visit and there was still hope for Mr Shaikh, she added.
"China does have a history of granting reprieves right at the last minute, literally as prisoners are walking out to the execution ground, so it's definitely not time to give up hope yet," she said.
Reprieve says Mr Shaikh was visibly ill to those who knew him in Poland
The charity also released witness statements it said it had collected from people who knew Mr Shaikh while he was living homeless in Poland.
Reprieve's witnesses all say Mr Shaikh had been clearly mentally ill when they knew him, and that he was fixated on recording a song that he believed would usher in world peace.
The Chinese authorities said they would notify Mr Shaikh about his execution 24 hours beforehand, but it is not certain if they have done so - aside from having informed his family.
His cousins also handed in a petition to the local court asking for a stay of execution and a mercy plea to the Chinese President Hu Jintao.
They were the first family members to have face-to-face contact in two years and the cousins are now on their way back to the Chinese capital Beijing.
'Erratic and extreme'
Mr Shaikh's family and the UK government have been calling for clemency but Mr Shaikh's final appeal was turned down last week.
He is set to become the first EU national to be executed in China in 50 years. The last Briton executed abroad was John "Jackie" Elliott, 42, who died by lethal injection in Huntsville, Texas, in February 2003 having been convicted of rape and murder.
His family and his defence team have said Mr Shaikh suffers from bipolar disorder and did not know what he was doing.
They say he was duped by a criminal gang into carrying a suitcase that did not belong to him.
His daughter Leilla Horsnell has said he was approached by drug smugglers in Poland and they convinced him they would make him a popstar in China.
CHINA DEATH PENALTY
China executed 1,718 people in 2008, according to Amnesty International
Last year 72% of the world's total executions took place in China, the charity estimates
It applies to 60 offences, including non-violent crimes such as tax fraud and embezzlement
Those sentenced to death are usually shot, but some provinces are introducing lethal injections
She has spoken out about his mental health problems and his "erratic and extreme" behaviour.
She said she hoped the authorities would listen to the family's final pleas but admitted she was not optimistic "because they haven't looked at the evidence previously".
"I'd like to be hopeful, but time just seems to be running out," she added.
MDF, a charity for people coping with bipolar issues, said it had written to the Chinese ambassador to add its voice to pleas for clemency.
Mental health charity Sane said the Chinese authorities "may have overlooked" the impact of bipolar disorder on a person's behaviour.
A spokesman for the Chinese embassy in London said Mr Shaikh was found with more than 4kg of heroin, which he said was enough to kill 26,800 people.
"Drug trafficking is a grave crime worldwide," he said. "The general public has a deep-seated hatred toward it."
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