Foakes did not take the field on the third afternoon against New Zealand after complaining of back pain, but later returned a positive test after a check up at the team hotel in Leeds.
Jonny Bairstow deputised with the gloves during the tourists’ second innings but Billings, who made his Test debut in the final match of the Ashes, has been drafted to take over for the rest of the match.
Under the new rules which allow substitutions for certain ailments, Billings will be allowed to bat in England's second innings.
A statement from the England and Wales Cricket Board read: “Ben Foakes was unable to keep wicket on day three of the Third LV= Insurance Test at Headingley suffering from back stiffness. On a further medical assessment yesterday evening, he subsequently tested positive after administering an LFT COVID-19 test.
“Details of his return to the England set-up will be announced in due course. However, it is hoped he will be fit for the LV= Insurance Test against India starting next Friday at Edgbaston.
“The rest of the England party follows health protocols of symptom reporting and subsequent testing if required. There are no other positive cases in the camp.”
Billings becomes the second substitute to play for England in this series, with Lancashire leg-spinner Matt Parkinson handed a debut at Lord’s after Jack Leach fell awkwardly in the field.
Foakes is the second member of the England set-up to contract Covid in the past week, with batting coach Marcus Trescothick unable to join up with the team for the series finale after a positive test.
New Zealand have also been hit, with captain Kane Williamson ruled out on the eve of the second Test at Trent Bridge. Devon Conway, Henry Nicholls, Michael Bracewell and two members of support staff have also had the virus since arriving in England.
And India have a Covid concern of their own after captain Rohit Sharma tested positive during their tour match against Leicestershire on Saturday.