Sincere thanks to everyone who replied off-list. It appears, sadly, that the recordings have not survived into the 21st century.

2017-08-29 12:59 GMT-04:00 Andrew Ollett <andrew.ollett@gmail.com>:
Dear list members,

I've come across a reference, in Alaka Atreya Chudal's book on Rahul Sankrityayan, to recordings of Sanskrit and Prakrit texts (Arthaśāstra, Nalopākhyāna, Setubandha, etc.) that the Mahapandit made while he was a professor at Leningrad University (1945-1947). These recordings were evidently available some decades afterwards in "the acoustics room at the university," according to V. I. Kalianov. I've written to the institutions at St. Petersburg that might know of the current whereabouts of these recordings, but I thought that I would ask all of you if you have any further information, especially since Sankrityayan is a relatively frequent topic of conversation on this list. It would be great to put those recordings online.

I should also take the opportunity to ask whether any of you are aware of any other recordings of Prakrit literature. I was surprised to see the Setubandha in the list, since I know of many fine recordings of classical Sanskrit and Kannada texts, but haven't yet found anything comparable for Prakrit texts.

Andrew