My suspicion is that, given the obviously poor management on this ward, the staff have become lazy and ill-disciplined. Too much hanging about chatting, making up names for people, not enough of doing the right kind of work.
The student is up-to-date in best practice, in theoretical terms at least, so notices all of this and knows how things ought to be done better.
The problem is in the way she expresses that, who she expresses it to and what happens when she does. That is, when she's raised issues with the manager, nothing has changed. Whereas when she pulls up individual team members, they jump to her tune (presumably because they know they're doing / not doing the right thing and feel guilted into action).
Hence all hanging around the lunches when there was other, higher priority work to be done. Because if it wasn't time for the ward round and that wasn't a higher priority, the nurse she'd pulled up would have said 'it isn't time for the ward round, we do that after lunch, as you know', or similar.
The problem is management. The student is right but also annoying.
So what should you do OP?
Distance yourself from the bullying. You're dead right to feel uncomfortable about that. Say out loud to colleagues that you're not happy about the nickname and mug and that it feels uncomfortably like bullying to you.
Keep a diary of incidents. Write down that you've said the above, the date and to whom, for example. Write down every issue you've raised with managers, date and their response.
Keep reporting issues to managers, for them to deal with. Only real issues, not gossip or opinions. It doesn't matter whether they feel they can bothered to do their job or not. It certainly isn't your job to do it for them. So you have no choice but to pass issues up the chain.
Look for another job in a better team. At some point, shit is going to hit fan in this one and the fallout may be messy and unpleasant.