Either it spread almost instantly and saturated whole populations, or else it is spreading at a finite and measurable rate. You cannot have it both ways.
Not trying to, it spreads at a finite, measurable but rapid rate. It has a long incubation period, and a long duration until the virus is cleared. Most people remain infected for several weeks, with no or mild symptoms, but the small percentage who develop serious symptoms do not happen in a day, it takes time. Thus by the time you are aware you have a problem that is not just ordinary flu and pneumonia, it has already spread through a large part of the population.