Is it acceptable to write a report for one university mentioning unpublished research done in another?
I don't understand the basis of your worry unless you signed a non-disclosure agreement as part of your internship. Don't assume that everyone hearing of your research, even in some detail, will immediately want to (or could) try to scoop you on a publication. People generally are more honest than that. 

But it is, I think, fairly easy to discuss research without giving up too much in the way of detail on the conclusions or how they were reached. General statements about the field, the methodology, and your part in the larger enterprise are probably sufficient in any such report. End it with the statement that the results are being prepared for publication. 

People in academia share quite a lot with colleagues. They even ask for and get advice without having to be paranoid about primacy or getting scooped. Getting scooped normally happens when two groups work on the same problems in parallel, often without knowledge of one another, but occasionally in competition.