Anderson and Wood missed the New Zealand tour due to injury, while Bairstow was dropped to allow him time to work on his red-ball game. All three are already in South Africa as part of a training squad.
While Wood is not currently deemed fit for selection, the England management hope he will recover sufficiently to play some part in the series. He was player of the match in the only Test he played in 2019, England's victory in St Lucia.
But there is no recall for Moeen Ali. Moeen, who requested an extended break from the Test team after struggling during the English summer, has held discussions with the England management in recent days with the conclusion that he should be given a little longer away from the international game. He is expected to return to the Test squad for the tour of Sri Lanka, where England may well field three spinners.
Moeen's absence allows Matt Parkinson, who went on the New Zealand tour but did not break into the Test team, another opportunity. Zak Crawley, who made his Test debut in the Hamilton Test, also retains his place as a reserve batsman. That means there is no place for Dawid Malan.
The return of Anderson and Wood means no place for Lancashire seamer Saqib Mahmood. He is the one man from the Test squad in New Zealand to miss out.
"With James Anderson and Mark Wood returning to the squad, Saqib Mahmood is the unfortunate player to miss out," said National Selector Ed Smith.
"It is not predicted that Mark Wood will be available for selection for the earliest matches. However, he will travel with the team and continue his rehab work with the medical staff on tour. There is no exact date given for his return, but the medical team is working towards him becoming fully available for selection during the tour.
"With that in mind, and given the extended period that James and Mark have had on the sidelines due to injury, it was sensible to have a 17-man squad.
"Moeen Ali, who asked to take a break from Test cricket last summer, remains unavailable for Test selection.
"Jonny Bairstow returns to the England squad after missing out on the New Zealand tour. Jonny, Mark and James are already in South Africa on a training camp ahead of the Test tour."
The four-match Test series forms part of the ICC World Test Championship and will be the first time in the competition that England have played away from home. The first Test begins on Boxing Day at Centurion.