I have studied at a similar level.
I did bump into a client there.
Tutors and other students on the course did know (inconfidence) about my paid work.
I didn't put specific details on my website about the uni or the course I was on.
Nothing bad happened, most people just got on with their own lives.
I did find it difficult to fit in studies around clients, sometimes new clients would try and persuade me to see them even though I'd set time aside for studies.
My advice would be to be strict with your time.
I also found it difficult to answer emails and phones, texts at the same time. I learned to not take my 'work' phone to uni, I have heard of others managing fine but I found it too distracting - I couldn't be all there if I was trying to arrange bookings.
Most regular clients and some new clients fitted into my new schedule, new commitments very easily and many were supportive. The others who were not tended to melt away.
Sometimes clients would expect an immediate answer to texts or emails, and I'd try and answer in snippets of time I had straight after lectures/uni. I also found this was not good, it increased the pressure and enquiries were not always happy if I said I couldn't see them (on an advertised day off).
For me sectioning plenty of time off I found worked best.