The government says it has signed contracts reserving 4,600 rooms in 16 hotels near English airports.
However, Health Secretary Matt Hancock said there is "some way to go" before lockdown can be eased, stressing the government was awaiting key data on how successfully the vaccines reduce transmission.
The new quarantine policy requires all United Kingdom citizens and permanent residents entering England from 33 "red list" countries to self-isolate at their own expense in approved hotels for 10 days and take several Covid-19 tests.
Arrivals found to have given false information about being in one of the countries 10 days before travel could receive up to 10 years in prison - which has drawn criticism for being excessive.
The government says it has signed contracts with 16 hotels so far, securing almost 5,000 rooms near English airports, with a further 58,000 rooms on standby, as it belatedly follows the example of others such as Australia and New Zealand.
Hancock said the government had resolved initial problems with the new policy at Heathrow Airport, Britain's busiest aviation hub.
"We've worked through all those issues with them".
Unless exempt, a passenger will have to pay £1,750 (€2,003) to quarantine in a room at one of six designated hotels in a bid to avoid importation of the virus.
The plane from Istanbul had three passengers rather than the 60 who were originally booked.
There is a £1,000 penalty for any global arrival who fails to take a mandatory test and a £2,000 penalty for any worldwide arrival who fails to take A second mandatory test.
People arriving in Britain must quarantine in hotels starting Monday as the government tries to prevent new variants of the coronavirus derailing its fast-moving vaccination drive.
On Sunday the government reached its goal of giving the first of two doses of vaccine to 15 million of the most vulnerable people, including health care workers and over-70s.
Ministers last month scrapped the previous "travel corridors" system that allowed unchecked travel from countries with lower caseloads.
But now the Scottish government says on the gov.scot website: "If you are travelling from outside of the Common Travel Area, and you do not qualify under an exemption, you will be required to quarantine in managed isolation for 10 days on arrival in Scotland".
Speaking at a health clinic in southeast London, Johnson said the government needed to be "very prudent" as it reviewed a third stay-at-home order in England that has shut down schools, non-essential businesses and hospitality venues since early January.
Charlie Islam-Harry, manager of the St Giles Heathrow hotel, said staff would do regular welfare checks on guests during their enforced stays, adding: "It can't be easy for them to be in a bedroom for 10 days".