The home team were forced into changes after the first Test victory at Lord's with regular captain Ben Stokes and Gus Atkinson dropped due to disciplinary issues while seamer Ollie Robinson has a knee injury.
Cox replaces Stokes at number 7 in the batting order and fast bowler Baker will come in for Atkinson, while Jofra Archer also comes into the starting line-up after being rested for the Lord's Test.
England have have also decided to drop specialist spinner Shoaib Bashir with Matt Fisher, who has one previous Test appearance, completing an all-pace attack.
Interim captain Joe Root and Jacob Bethell are expected to take spin duties if required. Cox leapfrogs James Rew straigt into the XI after the latter was selected in the squad ahead of the first Test.
England head coach Brendon McCullum said: "James Rew was our back-up batter in the squad, but that was very much to cover the top-six role. Jordan Cox, he obviously was over at the IPL. We wanted to see him come back and see what he could do before we thrust him into that role for the first time.
"We still very much see James Rew as an exciting option for us - and he may even get the nod this week, too - but Coxy's a fine player. He's been on our radar for a long time across all formats."
Baker struggled struggled on his ODI and T20I debuts last year but received a ringing McCullum is in no doubt that the paceman will be fine international cricket.
"He's got good air-speed, he swings the ball, he's got great skills," McCullum said. "But he charges in and he's got wild celebrations, and you can just see that cricket is what he wants to do.
"I'm excited to see how he goes and he is very well-planned and very well-prepared, which fits in with what we're trying to achieve now in this team with some of the shifts we've made."
England XI: 1 Emilio Gay, 2 Ben Duckett, 3 Jacob Bethell, 4 Joe Root (capt), 5 Harry Brook, 6 Jamie Smith (wk)*, 7 Jordan Cox, 8 Jofra Archer, 9 Josh Tongue, 10 Matt Fisher, 11 Sonny Baker
*James Rew to replace Smith if unavailable on paternity leave.
England have added seamer Henry Crocombe to their squad ahead of the second Test against New Zealand after Ollie Robinson experienced knee soreness in training and is set to have a scan.
Former England captain Michael Vaughan believes it's an overreaction to remove Ben Stokes as Test captain for breaking curfew.
Joe Root will stand in as England captain for the second Test against New Zealand after Ben Stokes and Gus Atkinson were made unavailable following a breach of team protocols.
Ben Stokes' could be under pressure to be retained as England Test captain after it was revealed that he and teammate Gus Atkinson breached team protocols following the victory against New Zealand at Lord's.
