Pre-departure Covid-19 testing will now be required for everyone travelling to England from 04:00 GMT on Monday.
The rules had been due to come into force on Friday, but the government said people needed time "to prepare".
Those arriving by plane, train or boat, including UK nationals, will have to take a test up to 72 hours before leaving the country they are in.
Anyone arriving from places not on the UK's travel corridor list must still self-isolate for 10 days.
The Scottish government is planning to impose the same rules and has had to defer them coming into effect as a result of changes in England.
"This meant Scotland was also obliged to delay implementation as we need sight of their final regulations in order to properly draft and approve the relevant Scottish regulations," a spokeswoman said.
It is expected the requirement will come into force in Scotland at 04:00 GMT on Monday as well. Wales and Northern Ireland are expected to announce plans for pre-arrival testing in the coming days.
Announcing the deferral on Twitter, Transport Secretary Mr Shapps said: "To give international arrivals time to prepare, passengers will be required to provide proof of a negative Covid-19 test before departure to England from Monday 18 January at 4am."
He also reminded travellers to fill out the Passenger Locator Form - used in track and trace - and added that those without proof of a negative test faced a fine of £500.
Problems with testing availability and capacity mean some countries will initially be exempt.
For instance, the requirement will not apply to travellers from St Lucia, Barbados, Antigua and Barbuda until 04:00 GMT on 21 January.
Travellers from Falkland Islands, Ascension Islands and St Helena are exempted permanently.
Hauliers are exempt to allow the free flow of freight, as are air, international rail and maritime crew.
The government has said all forms of PCR test will be accepted, as will other forms of test with "97% specificity, 80% sensitivity".
New strain in Brazil
The move comes as a further 1,564 people have died in the UK within 28 days of a positive Covid test - the biggest figure reported in a single day since the pandemic began.
Wednesday's figure brings the total number of deaths by that measure to 84,767.
Dr Yvonne Doyle, medical director at Public Health England, said there had now been more deaths in the second wave than the first.
Meanwhile on Wednesday, Prime Minister Boris Johnson said he was "concerned" about a new coronavirus variant that is believed to have emerged in Brazil.
He acknowledged it was not yet clear how effective existing vaccines would be against the latest new variant.
Mr Johnson said the UK was taking steps to make sure it was not brought into the country.
A government Covid committee is meeting on Thursday to discuss the possibility of stopping flights from Brazil.
Arrivals from Brazil already have to self-isolate for 10 days.
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