Neither particularly likely but both interview significant percentages of applicants. Completely at a guess i would say Cambridge seem more liberal and may be more likely to view your case kindly. but bear in mind views on what are extenuating circumstances vary depending on your illness, your college will probably not be much of a factor but i could be wrong. Other people have issues too, you should just ask the university what they think.
* few people from your school/college proceed to higher education ,,i was wrong :P AND your family has little or no tradition of studying for a degree
* your education has been significantly disrupted or disadvantaged through health or personal problems, disability or difficulties with schooling