On Tuesday, Dewani launched an attempt to take his case to the Supreme Court after a judge ruled he should be extradited.
It is the latest bid to stop the businessman being sent back to face trial over the murder of his bride in a township outside Cape Town in November 2010.
Lawyers for Dewani, 33, who has suffered from depression and post-traumatic stress disorder since the killing of his Swedish wife Anni, had argued that his mental health is too fragile.
Chief Magistrate Howard Riddle gave the go-ahead to Dewani's extradition in 2011, but had to reconsider the position after two senior High Court judges allowed an appeal in March last year.
Riddle ruled last month that Dewani should be extradited.
Dewani's lawyers lodged an application at the High Court for the two judges there to certify that their ruling raised "a point of law of general public importance" for consideration by the Supreme Court, the highest court in Britain.
John Thomas and Duncan Ouseley said it would be "unjust and oppressive" to remove him until he recovered -- but it was plainly in the interests of justice that he was extradited as soon as he was fit.
Anni Dewani, who like her husband was of Indian origin, was shot dead when the couple's vehicle was apparently carjacked.
Dewani denies any involvement in the killing.
A South African man, Xolile Mngeni, was jailed for life for the murder last December. Two other local men jailed over the killing allege that Dewani ordered the hit.
South African authorities have been seeking Dewani's extradition since December 2010.
He was arrested in Britain and released on bail, but British courts have delayed the extradition until now on the grounds of Dewani's serious mental illness.
Dewani has spent more than two years on mental health wards -- a far cry from his previous life as a wealthy executive for his family's chain of nursing homes for the elderly.