The Help to Buy Mortgage Guarantee Scheme allows lenders to offer loans to people offering a deposit as low as 5% thanks to government support.
The scheme is set to be closed at the end of the year.
Use of the scheme has declined, while mortgage lenders have reported that business in general is up sharply.
Lending to people with a 5% deposit, at a similar if not cheaper price, is available outside of the scheme.
In a letter to Chancellor Philip Hammond, the Bank's governor Mark Carney said that closing the scheme as planned would not cause lending to dry up.
"Given the decreasing usage of the scheme over time, the [Financial Policy] Committee judges that the closure of the scheme would be unlikely, in current market conditions, to affect significantly the provision of finance to prospective mortgagors, including high loan-to-value borrowers," he wrote.
The move is unlikely to affect other Help to Buy schemes, which include equity loans to buy newly-built homes, as well as the Help to Buy Isa savings scheme which is aimed at those saving for a deposit.
In its review of the Help to Buy Mortgage Guarantee Scheme, the Financial Policy Committee (FPC) said that the scheme accounted for 3% of total mortgage lending in the first quarter of the year. In 2014, it accounted for 6% of such lending.
It accounted for 25% of lending in the first three months of the year to those offering a deposit of less than 10% of the value of the home to be bought, compared with 70% in 2014.
There had been fears when launched that the scheme would create an unstable housing market bubble, but the FPC said that it had not posed a risk to financial stability.
It added that the scheme had not driven up prices in general, nor on the houses bought through the scheme.
There was a cap on loans which meant borrowers were only allowed up to 4.5 times their income. This, said the committee, had prevented a long list of highly indebted borrowers from building up.
The health of the mortgage market was stronger in August, separate figures from lenders have revealed.
Mohammad Jamei, senior economist for the Council of Mortgage Lenders (CML), said: "Widely voiced fears in recent months about the housing market have proved to be wide of the mark."
He said that there had been a post-referendum lull, but lending had rebounded in August, partly as a result of stimulus from the Bank of England - including a cut in interest rates.
The CML said that gross mortgage lending in the UK totalled Â£22.5bn in August. This was 7% higher than the previous month and 15% higher than the same month a year earlier.
It was the highest level of lending during any August since 2007, the CML said.
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It is understood the plane came down at the privately-owned Peterborough Business Airport at about 11:50 GMT.
The pilot, a man aged in his 40s, was taken from the airfield to Peterborough City Hospital with a serious leg injury.
An air ambulance and other emergency vehicles were sent to the scene.

With just one game remaining and only the group winners to qualify, Nigeria cannot make up their five-point deficit to table-toppers Egypt.
Ramadan Sobhy's 65th-minute winner puts Egypt on the verge of qualification.
Egypt face Tanzania in June and will book their place in Gabon with any result better than a 3-0 defeat.
Nigeria won the Nations Cup in 2013 but have now suffered back-to-back eliminations in qualifying.
They threw everything forward to find an equaliser on Tuesday and came close when West Ham winger Victor Moses crashed a shot against the post in the 84th minute.
The match in Alexandria became a virtually all-or-nothing tie for Nigeria after Chad withdrew from the group on Sunday, citing financial difficulties, and all results from their matches were erased.
That left only three teams in Group G and in accordance with the rules of the Confederation of African Football only the winner would qualify for the finals.
Egypt have seven points with one match to play, against bottom club Tanzania, who have only one point but two games remaining.
To have any chance of qualification Tanzania would need to beat Egypt by a better scoreline than the 3-0 defeat they suffered in Egypt last June because head-to-head records would come into effect if the sides finished level on points.
Tanzania would still have to beat Nigeria in their final match in September.
A major scheme designed to prop up the UK housing market is no longer needed, the Bank of England has told the government.
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A pilot has been injured in a light aircraft crash.
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Nigeria have failed to qualify for the 2017 Africa Cup of Nations after falling to a 1-0 defeat in Egypt in their Group G match on Tuesday.