Both universities run a jobs board for students. Register in the first week of term. Campus roles go quickly, and the board is a better start than a generic website.
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Indeed and Totaljobs are worth a look with the part-time filter on. For cafes, shops and bars, walking in on a weekday afternoon with a one-page CV still works, particularly in Hockley and the Lace Market. Go when they are not serving lunch.
If you are on a Student visa, the usual limit is 20 hours a week during term, and full-time in the vacation. Check the vignette or the BRP decision. Do not take a number from a blog, including this one, over the wording on your own visa. UK and Irish students do not have that cap. A limit of your own, around 15 hours, is still sensible if the course is full-time.
Keep a note of where you have applied. A page in a notebook is enough.