Countries across Europe began banning flights from Britain after the disclosure the country is the centre of the outbreak of the new variant, which is up to 70 percent more transmissible than the original.
Concerns about the rapid spread of the disease were underlined with the publication of the latest official figures showing there had been a further 35,928 lab-confirmed cases of coronavirus in the UK as of 9am on Sunday.
Dr Yvonne Doyle, medical director at Public Health England, said: "This sharp and sudden increase is of serious concern."
She said most of the new cases were concentrated in London and the South East - where the new strain is thought to have originated - although it was too soon to say if they were linked to it.
Meanwhile, Ireland and France have become the latest countries to ban travel from Britain with the measures set to come force at midnight and last for at least 48 hours.
In Ireland, urgent talks were held on Sunday. In a statement afterwards, the government announced that flights arriving from the UK would be banned for 48 hours at least from midnight, and "in the interests of public health, people in Britain, regardless of nationality, should not travel to Ireland, by air or sea".
France has suspended all travel links with the UK for 48 hours from midnight. It said the period should be used to provide a co-ordinated European response.
In Germany, an order from the ministry of transport said planes from the UK would not be allowed to land after midnight on Sunday, although cargo would be an exception.
Earlier the Netherlands said it was stopping flights from the UK at least until the end of the year while Belgium has imposed a 24-hour ban on flights and rail links while it assesses the situation.
Italy is prohibiting entry to the country by anyone who had been in the UK in the last 14 days while flights are banned until January 6.
Austria and the Czech Republic are also imposing new restrictions, with Prague announcing stricter quarantine rules with anyone arriving in the country having spent at least 24 hours in UK territory required to self-isolate.
The moves come after Boris Johnson effectively cancelled Christmas for millions of families in London and the South East with a two-week lockdown over the festive period, with people limited to mixing with one other person from another household outdoors.
Wales has also begun another two-week lockdown while Scotland is to maintain a travel ban with other parts of the UK over the holiday period.
‘Out of control’
Mr Hancock said the country was facing an "enormous challenge" after scientists warned the new VUI 202012/01 variant could be up to 70 percent more transmissible than the original virus.
"The new variant is out of control and we need to bring it under control," he told BBC1's The Andrew Marr Show.
"We don't know how long these measures are going to be in place. It may be for some time until we can get the vaccine going."