People 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.
All those arriving from places not on the government's travel corridor list must still self-isolate for 10 days.
There are issues with testing availability and capacity, so some countries will initially be exempt.
For instance, the requirement will not apply to travellers from St Lucia, Barbados, Antigua and Barbuda for six days.