Hi all,
Just in case anyone is still following this thread: the Christadelphian Hall does seem to be same building as the earlier Hall of Sciences.
You can see on this map from 1890 the Hall of Science is located next to the Exchange Works: http://www.sheffieldarchivemaps.co.uk/strip_maps.asp?x1=435057&y1=386924&chk=1&buffer=11000&myscale=11000&zoom=0&mylock=1&sterm=ROCKINGHAM%20STREET&myArea=Sharrow
Then looking at this picture of the Exchange Works, its shown next to the Christadelphian Hall https://www.picturesheffield.com/frontend.php?keywords=Ref_No_increment;EQUALS;s19085&pos=2&action=zoom
Finally, there's a better (colour) photo of the hall from 1930 on the sheffield christadelphian website here http://www.sheffieldchristadelphians.org.uk/about-us
So they definitely seem to be the same building, but it would be great to see what it looked like as the Hall of Science!