Within 24-48 hours you'd start feeling the effects of skin conditions from unchecked staphylococci and the like, such as eczema and erythrodermatitis.
It is difficult to come up with a hard, reliable source for infection progression since instantaneous immune collapse isn't really a thing. However, ex novo infection can have an incubation as short as 96 hours and in-vitro growth time is reported as about one day (google 'staphylococcus growth time' or 'incubation period').
As soon as the skin barrier is broken, cellulitis and circulation problems will ensue, followed (since there's nothing to check the infection) by gangrene. Within three days you will be severely disabled and within four or five days pretty much be unable to move. Expect death from septic shock to occur within one week. One week, in this case, refers to a time period not exceeding 7 Earth-days.
Time-frame data is also difficult to obtain. I have used data from several instances of severe Clostridium infections, where "symptoms usually develop six to 48 hours after the initial infection and progress very quickly" in compromised subjects.
- 0 - 24 hours: No real effects except itching and rashes
- 24 - 48 hours: Weeping sores, discoloration and loss of sensation in limbs and other areas. Some loss of functionality may be noticeable after healing.
- 48 - 72 hours: Severe pain and gangrene. Even after immune system recovery, there are strong chances of amputation being required, scarring and permanent loss of some functionality.
- 72 - 96 hours: Almost certain amputation necessary, possible death by septic shock even after healing.
So, while in that time several nasty kinds of cancer are guaranteed to have sprung up, they won't be what kills you.
(If several brief immune loss periods repeat, however, it's possible for some of those cancers to develop enough to become dangerous).
Meanwhile, however, dormant infections might spring up and manifest (herpes, for example, and some fungal conditions). These will take longer to disappear even after healing (actually might surface some time after the immune system has recovered).
If you have a preexisting condition or eat anything that requires immune support (the bactericidal properties of saliva will still be there, but several kinds of spores will not be killed and usually die when they germinate and get recognized by the immune system - that barrier will no longer hold), then anything from cholera to Montezuma's Revenge can kill you within 48 hours.
If you don't have complete immune deficiency, then you can survive indefinitely as long as you take sufficient precautions (this is the so-called "Bubble Condition" or "Bubble Boy Disease").
Make it more severe than that and you have something not too different from Acquired Immunodeficiency Syndrome - AIDS.