Is it acceptable to write a report for one university mentioning unpublished research done in another? Last year I got in touch with a professor by email. I was somewhat familiar with their topic, and very enthusiastic about working with them. I had a very positive exchange with them and did a call, after which they were willing to offer me a research internship. Their university is in the US, the field is computer science, and I am an undergrad studying outside the US. 

The internship went well as far as I could tell. At the end, they offered to write me a recommendation letter (without me bringing it up), and we agreed to keep working remotely after I returned home. We had a very interesting discussion going on, even after the official internship ended, which got to the point where they asked me about my intentions for graduate school. By that time I was also finishing up the results of the work I did, in preparation for publishing it at a conference, which we had talked about when I was still in the US. It was to be my first publication.

Back at my university, I was asked to write a report of the work I did, because it was counted as the mandatory internship experience required to complete my undergraduate degree. I knew beforehand that I was to write a report, but took it as administrative overhead which nobody takes very seriously, and did not mention it to my internship supervisor until after the internship was over.

I sent them an email before writing the report, saying that I couldn't write anything very convincing about the work I did without at least mentioning some of the ideas we had discussed. I asked them about the level of detail they would be comfortable with me including in the report, given that we intended to publish. I also mentioned that the report would be used by my department (which is Electrical Engineering) only for evaluation purposes, and that it would not be read by anyone working in a field remotely close to the one we had worked on during the internship.

After that email, I did not hear back from my internship supervisor at all. Many months have passed now. I even sent them the results of the internship work, and multiple reminder emails.

Was it wrong to fail to mention that I had to write an evaluation report earlier? Or was it just unacceptable to write a report at all about unpublished research? Might they have felt that my email was not written in good faith and that I was trying to share their ideas with my department, which in reality hadn't the slightest idea of the topic I was working on?
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.