For 66 years, the two spools remained unaired and unheard—safely locked up in the vaults of the archives of All India Radio in India’s capital. The labels on them are terse: ‘Jinnah’, they read. Only a few knew of their contents or value. Unlike other recordings of historical import, neatly marked for date, time and context, the ‘Jinnah tapes’ were something that existed behind a bureaucratic iron curtain.

Now, on hearing the master tapes after an eight-month pursuit, this much can be said: Mohammed Ali Jinnah sounds every bit like the man he despised—Pandit Jawaharlal Nehru. Only, the accent is stiffer, more gravelly, more Brit. And yes, in one of them at least, the Karachi-born barrister makes the same noises as the one from Allahabad.

The first of the two recordings we heard was perhaps Jinnah’s last address on the radio within the borders of what is now India. It was made on June 3, 1947, in Delhi, two months before he left for the country that had become his life’s mission. All India Radio’s internal...