Du Plessis has stayed in South Africa to be with wife Imari following the birth of the couple's first child.
"Faf's wife gave birth to their first child last week but the difficult nature of the delivery has delayed his return to the UK," team manager Mohammed Moosajee said in a statement.
"The mother and baby have settled at home, but he will be unable to make it back in time to take part in the first test. He will link up with the squad in the UK later this week."
Theunis de Bruyn or the uncapped Aiden Markram will replace du Plessis' in the side.
Opening batsman Dean Elgar will lead the team and become the 12th player to captain South Africa since their readmission to international cricket in 1992.
Better news for the squad is that experienced seamer Vernon Philander has been cleared to play.
He had sat out the drawn warm-up fixture against England Lions with an ankle injury.
(Reporting By Nick Said; Editing by Amlan Chakraborty)
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