The US is to airlift its citizens from the Diamond Princess, which has reported 285 cases of the coronavirus – the biggest cluster outside China.
In a video on Facebook, David Abel said he had little hope of a similar rescue.
The Foreign Office told the Observer it was “working around the clock” to ensure the welfare of Britons onboard.
But Mr Abel, who describes himself as a “staunch Tory”, said he had “no confidence” in Prime Minister Boris Johnson.
Speaking during a Facebook live video with his wife, Sally, he said: “When he [Johnson] just says ‘keep calm, don’t panic’ – I’d like to see you in this situation, mate. I really would.”
Mr Abel, who has become an unlikely celebrity as a result of his videos, went on to appeal to billionaire Richard Branson for help.
“So, Richard Branson, I want to ask you a question, pal,” he said in a video posted on Valentine’s Day.
“If you and your family were in this situation, what would you do? And please don’t say ‘chill out, stay calm’, that’s not what we want to hear.
“I’m asking, what would it cost to hire one of your smaller planes, put all the Brits on board, no flight attendants, packaged food?
“Take us to Brize Norton, take us straight into the medical facility and let us do our quarantine there by people who can speak our language.”
Mr Abel added that his appeal was a reflection of “just how desperate some of the passengers are becoming”.
He and his wife have previously said they have been given the option to leave and continue their quarantine ashore if they test negative for the virus, but have chosen to remain on board.
On Saturday, NHS England said all but one of the nine people being treated for the coronavirus in the UK have been discharged from hospital.
Elsewhere, the first coronavirus death outside of Asia was confirmed as having happened in France.
The victim was an 80-year-old man from China’s Hubei province, according to French Health Minister Agnès Buzyn.
On Sunday, China announced a drop in new cases of coronavirus for a third consecutive day.
In total more than 68,000 people have been infected in China, with the death toll at 1,665.
Outside of China there have been more than 500 cases in nearly 30 countries. Four people have died – in France, Hong Kong, the Philippines and Japan.