Having looked through Bmd records from the 1830's onwards, no one called Paternoster has ever been connected with Sheffield, so it probably writes that theory off.
Paternoster Row predates the Paternoster lift by half a century.
A possible connection is printing and publishing...a search of the newspaper archives using 'Paternoster Row' shows that from 1750 every reference for many years is around books and publications over a wide number of towns and cities, are supplied from Paternoster Row, London, although binding may be done in the provinces.
By the 1800's Wolstenholme's academy of learning has sprung up on Paternoster Row, Sheffield, in the 1830's Baldwin & Cradock of Paternoster Row are advertising as suppliers of a large range of books.
Not definitive at this stage but possible that the street was named after the London street.