Only months ago, India ran so short of Covid-19 vaccines that computer experts wrote software code to help people snag scarce immunisation slots and the then health minister constantly battled opposition criticism on social media.

The minister later quit and in came a little-known but hands-on politician, Mansukh Mandaviya, who, according to a person close to him, communicates with vaccine producers on a near-daily basis and tries to resolve their problems promptly.

Vaccine supplies have surged.

Government officials say they are...