Can the institution demand the exam scripts back?
Yes. The legal position is that the exam scripts/ essays/ other materials to be assessed are the property of the institution and the institution may demand their return (marked or non-marked) at any time. Remember all the union is asking you to do is not to mark or formally assess students’ work. The scripts or essays or other pieces of work to be assessed could have e.g. comments written on them but not marks. Students may be informally given general feedback but formal marks should not be recorded or conveyed.