Good question, it depends on the bug. The bacteria that cause tuberculosis (TB) often stay inside the lungs for years, even someone’s whole life, without ever causing disease.
On the other hand anthrax bacteria that live in the ground can spread within the body in less than a day and cause illness which can quickly lead to death without treatment. Because it can cause death so easily, it’s sometimes used as a weapon, with some cases of letters containing the bacteria being posted to people. Luckily we have a vaccine for people who think they might be exposed to anthrax and treatment for people who have been exposed.
There are lots of different types of bug, so there might be some that spread faster that I don’t know about.