Do you have a personal tutor? The best thing you could do is to get in touch with them and ask about the procedure. When I was at uni, we were told that if we were ill we should let them know - because if we come to the exam there's nothing we can do about it, we have to take our grade even if it's rubbish, whereas if you let them know in advance they may be able to do something about it.
Your uni might be different, but this is how it would have gone at mine.
Uni Admin offices will have been open from last Monday - they don't keep to student term terms. Contact them asap.
Most will allow you some kind of extenuating circumstances, but as you say, it will depend on the uni. At mine, if you were too ill to take the exam at all, you would normally be allowed to take them for the first time during the resit period. If you were well enough to sit the exam but had an illness likely to affect your performance, then you would be given additional marks to compensate for whatever the uni thinks your shortfall might be as a result. Both of these will require a doctor's note as proof, so get back to your GP pronto and ask for one.