Captain Ben Stokes and pace bowler Gus Atkinson have been made unavailable for the second Test following an incident in a nightclub following the first Test, while seamer Ollie Robinson is injured.
As a result of the absentees, and to balance the team, England have omitted spinner Shoaib Bashir from the XI that won the first Test.
In have come pace bowlers Jofra Archer, Matthew Fisher and Baker, alongside Cox and Rew.
It is the first time England have made five changes to their team during a home series since the fourth and final Test against New Zealand in 1999, also the last time they faced the Black Caps at The Oval.
Smith, 25, missed the tour of New Zealand in 2024 for the birth of his first child.
On that occasion, Cox was due to deputise, only to suffer a broken thumb in the nets.
Although Rew was named in the England squad for the first Test, Cox moved ahead of him in the pecking order for the second Test and was named in the hosts’ XI on Monday.
Rew, who averages almost 42 in first-class cricket and has made 12 centuries, spent a long period on Tuesday practising his wicketkeeping at The Oval with England fielding coach Sarah Taylor.
Despite Rew taking the gloves in the second Test, he will bat at number six, one place ahead of Cox.
Smith is the second player to welcome a child during this series.
New Zealand opener Devon Conway returned home after New Zealand’s defeat at Lord’s to be at the birth of a baby, but has since returned and will play in the second Test.
