Chasing a target of 175, England were on course of a series-clinching win with 133/2 in the 16th over bowled by Chris Morris when Roy, who was batting superbly on 67, was given out.
South Africa made 174 for eight with JJ Smuts contributing 45 and skipper AB de Villiers striking 46 from 30 balls.
The process has been now extended by 16 days, which effectively gives former Team Director Ravi Shastri a chance to throw his hat in the ring.
Bumrah, who plays for Mumbai Indians in the Indian Premier League, overstepped when he got the wicket of Fakhar Zaman.
The BCCI had decided to continue with Kumble for the Windies tour but veteran Indian spinner decided to call it quits.
India are in the group of death, which also has defending champions Australia, and New Zealand.
Pant, 19, is being groomed as a possible replacement for Dhoni, who is already 35.
Kumble resigned as the head coach on Indian team on Tuesday after days of speculation that not everything is right between him and Kohli. The former leggie shared with fans why took the step in a Twitter post, and since many have asked Kohli's version.
India have had an impressive build-up to the 50-over showpiece, beating South Africa by eights wickets in the final of the Quadrangular series recently.
Meanwhile, coach-less Indian cricket team are in the West Indies for a short limited-overs series. The first of the five ODIs will be played today at Queen’s Park Oval, Port of Spain. The ODI series will be followed by a lone T20I match.