Conway's place in the side was under threat, with a top score of just 22 in his last seven test innings.
He scored 76 in the first innings of the second test against India in Pune but followed that up with scores of 17, 4, 22.
Against England he has scored just 2,8, 11 and 0.
Will Young, the player of the series for the Black Caps in their historic 3-0 test series win over India in India last month, will likely replace Conway at the top of the order while Auckland's Mark Chapman has been added to the wider squad.
"Family comes first in this environment and we're all really excited for Dev and his wife Kim to welcome their first child," coach Gary Stead said.
England have an unassailable 2-nil lead in the three test series after comprehensive wins in Christchurch and Wellington.
The third test in Hamilton starts on Saturday.
