The paternoster at the Arts Tower in the University of Sheffield, UK (Universities web-page, BBC video) certainly used to be open to the public. These forum posts from 2012 and a number of more recent Tripadviser reviews suggest that this is still the case. The university also publishes a self guided tour (pdf) which includes the tower and its blurb does talk about the paternoster but also states that there is "Limited access to inside building".
Alternatively you could book yourself onto a public event inside the building. Monday to Fridays 15th to the 29th of June an architectural exhibition is taking place in the arts tower. Or you can view their full list of events. The university has also traditionally opened one building up to members of the public to tour in September each year as part of the heritage open days scheme, the arts tower opened in 2012 and the university didn't take part last year (2017). Weather or not they will take part in 2018 and which building will likely be known around mid July (Source). Obviously there is no guarantee that the paternoster will be running on any given day/time. But i'd image its most likely to be the case from around 9-5 during term time. The building does feature other normal lifts which may be in use instead - something which I would imagine is more likely going to be done for a late evening event or on a weekend/event out of term time. You could send an email and try and ask the exact opening times.
Finally, you could also visit on an open day with potential students, there are a handful of dates over the summer or the students union offer more regular tours to non-students. They note that "Although the campus tour will take you past most of the academic buildings, if you wish to visit a specific department and speak to a member of staff then you will need to contact them directly, at least two weeks prior to your visit to the University." you could ask if there tour includes the Arts Tower, or ask if they would be willing to while you are there. While these tours are open to the public, they will of course be designed for prospective students of the university.