The selectors have been forced into a dramatic overhaul of the team for The Oval.
Captain Ben Stokes and seamer Gus Atkinson are unavailable following their unauthorised night out in London last week and Ollie Robinson withdrew due to a knee injury.
Cox replaces Stokes at No.7, fresh from a knock of 204 for Essex in his first County Championship outing for 11 months, while Hampshire quick Baker bolsters the bowling attack alongside the returning Jofra Archer.
Spinner Shoaib Bashir makes way having not bowled an over in the series opener.
Joe Root will lead the side once again, more than four years after handing the baton to Stokes, and a fifth personnel change could be coming should Jamie Smith's partner give birth in the next couple of days.
That would mean Somerset's Rew added to the XI, batting at six and keeping wicket.
It represents a remarkable situation little more than a week on from a 115-run victory but McCullum attempted to frame the positives of his enforced reshuffle.
"There's been a few changes," he said with heavy understatement.
"Whilst there's been a bit of chaos and it's been difficult to navigate through the week…. I look at that line-up we have this week and it's super exciting. I'm quietly confident we're going to see some exciting cricket."
Baker came close to winning his cap at Lord's but was ultimately overlooked. Now, having endured difficult first outings in both white-ball formats last year, he has the chance to make it third time lucky in an England shirt.
"I genuinely believe that if Sonny's able to get some early success, if he's able to get an early wicket, I think the crowd's going to get behind him," said McCullum.
Cox was denied a possible Test debut in December 2024 when he fractured his thumb in training, circumstances which led to Jacob Bethell's emergence. Cox has spent the majority of the domestic season warming the bench at Royal Challengers Bengaluru in India but roared back into contention to leapfrog Rew with his stunning knock against Leicestershire.
Smith was not present at nets on Monday afternoon, spending time with his family awaiting the arrival of his second child.
ENGLAND (v New Zealand, second Test): Ben Duckett, Emilio Gay, Jacob Bethell, Joe Root (capt), Harry Brook, Jamie Smith (wk), Jordan Cox, Jofra Archer, Josh Tongue, Matthew Fisher, Sonny Baker.
