Vice-captain Pope picked up the gloves on day two of Sunday's warm-up game, but with only 14 players in Queenstown and some opting for extra time at the nets, England had to ask James Anderson to take part as an outfield player.
Anderson, 42, has been working as England's bowling consultant since retiring from Test cricket in July and will take part in the Indian Premier League auction from Sunday.
The injury is cruel for Cox, who had to be brought down by England assistant coach Jeetan Patel.
Cox has been a reserve keeper and batsman in the Test team since the summer and had already been told he would make his debut in place of Smith.
England will demand a replacement but Cox's injury is the latest episode in an ever-changing picture behind the stumps. Smith, 24, was named first choice at the start of the home summer after Jonny Bairstow and Ben Foakes dropped out.
A return for both seems unlikely. Phil Salt has played white-ball cricket for England, while Durham's Ollie Robinson has impressed in the County Championship.
20-year-old James Rew from Somerset is an exciting prospect and is currently with the England Lions in South Africa, alongside Matty Hurst from Lancashire, another 20-year-old.
After conceding a deficit of 113 runs As both sides fought on the first day, The Prime Minister's side posted a score of 313-5 after 66 overs, with Olly Stone the pick of the England bowlers with 3-53.
Only Joe Root (82* from 54 balls) and Ben Stokes (59 from 39) were in the hunt for 201 despite there being little time left. They topped single figures, with the pair reaching 111 for the fifth wicket.
Despite wickets falling, the tourists almost secured an unlikely victory, but the game ended with England 196-9 after 22 overs, five short of their target.
