Mr Nielsen, then 19, found the tape when he visited the studio, near BBC Broadcasting House, in 1967 and was allowed to take unwanted recordings that were due to be burned.
He never listened to his haul and simply threw them in a box, forgetting about them for the next five decades.
It wasn't until he read guitarist Keith Richards' autobiography two years ago that he realised the world-famous band had recorded at the studio.
It prompted the 67-year-old to root around the attic and discover the treasure trove.
The Rolling Stones are understood to be excited at the prospect of hearing the song again. A source close to the band said: "They remember that era very well, when they were first starting out and before they started having hits."
The tape is being sold by auction house Tracks on April 3, with a pre-sale estimate of 6000 ($12,765).