Failing to find any reference to this apart from in Scotland where they are called a turnpike stair, although I think that might just be Scots for what a Sassenach like me would call a "spiral staircase" rather than being mine specific. Maybe for shallow shafts prior to engines it could be used, but can't see it as practical for much depth?
"...Windmills were also tried but were too irregular. An early method of working was a circular shaft, with a turnpike stair, up which in the days of slavery, the coal was carried in baskets on the backs of women, the coal bearers of that period. By the introduction of the steam-engine, water, coal, and other material can now be raised from great depths..." "Historic Notices and Domestic History of the Parish of Shotts", by William Grossart, surgeon, 1880