Investigation of Academic Misconduct I'm not sure what to tag this question as, and I don't know if it should be on Law.SE.

If someone is being investigated for academic misconduct:


accused of taking photos of tests and sharing it online, which is explicitly disallowed by the professor
and during that investigation, there is no evidence of the tests on the person's cell/camera,
but there is evidence of other crimes (completely unrelated to the university) on their cell/camera (e.g., video of this person bullying a child),


can the university/investigators take any action?

I think no, because the investigators were only to investigate academic misconduct and nothing else. But also yes, because they have evidence of improper behaviour.
University personnel will investigate whether the rules and norms of the university are being upheld.  If an actual crime is suspected of having been committed, then in most cases they will contact the police or other authorities either in place of investigating or in parallel to their own investigation.  For certain crimes they are legally obligated to do so.


  I think no because the investigators were only to investigate academic misconduct and nothing else


Just because they are only there to investigate academic misconduct does not mean that they cannot report nonacademic misconduct to someone else.  If they found evidence of a literal crime, I don't know why they couldn't report it to the police.  (I'm not sure exactly what "Bullying a child" means, but if e.g. this included physical or sexual abuse I would certainly expect them to report it to the authorities.)  

I think you are asking whether there is any covenant of confidentiality when you turn over your phone to them.  That's a legal question and I'm not a lawyer, but unless explicitly agreed upon in advance I don't know why there would be.  If your partner accuses you of cheating on them, you deny it, and they say, "Well then show me your phone," and upon searching your phone they find a video of you stealing a car, then your partner can certainly report you to the police even though that was not what they were expecting to find.  To my mind, having your phone searched by university personnel is more like having it searched by your partner than by the police.  In particular, university personnel cannot compel you to turn over your phone to them.  They can ask you to, and there might be negative consequences for you if you don't (just like with your partner), but that's not the same thing at all.  

Furthermore: as far as I know, university personnel may act on information they find on your phone that is unrelated to the academic honesty issue in question but otherwise violates the university's code of conduct.  For instance if they found a video of you making racial slurs at a campus event, then that's not illegal but is (I hope) against their code of conduct, and so far as I know they could act on that.  However I do not know exactly where to draw the line.  For instance, if in an internet dating app you use language that would be disrespectful and inappropriate in university discourse, then presumably that is not their concern...but if that language involved another university official it might become more their concern, perhaps.  I'm really not sure.

Let me say finally that I've heard a few times on this site from students who have had their phones searched by university officials, but as a university official (faculty member) that's something that I have never done and would not think to do in almost any situation.  For instance, if during an exam I saw a student looking at their phone, then I would call attention to it, have them put their cell phone away, and pursue an academic honesty case against them.  But I would not ask to see their phone: since I tell students before the exam starts what they can and can't have out during the exam (in particular, no phones), looking at their phone during the exam is cheating no matter what, and the implication that they are breaking the rules to gain an unfair advantage is more than plausible whether I see what's on the phone or not.  (And what am I going to do, scream at the student to drop their phone immediately and run over to apprehend it before they can close an app?  This seems silly at best.)  I wonder though whether in other parts of the world academic culture is different: in the United States (and Canada; I spent 2.5 years as a postdoc there) at least, the idea that people do not have to casually turn over their personal property is quite strong.