It depends on the kind of leukemia. Acute leukemias progress quickly, and treatment is needed to start immediately (usually within a day or two of diagnosis.) Without treatment, acute leukemias could be fatal in weeks (less for some people... in rare instances, an acute leukemia may stay quiet for a couple of months before diagnosis.) As far as how their condition changes over time, what do you mean? Without treatment? I'd imagine that as the bone marrow fills with blasts, the person would have drops in platelets (causing bleeding) and red blood cells (causing fatigue and shortness of breath) and good white blood cells (leading to infection.) They'd likely have bone pain, enlarged spleen and possibly liver, and a multitude of other symptoms. Thankfully, the only time I've seen a person get that far with untreated leukemia has been in a person who relapsed multiple times and treatment was no longer an option.
Chronic leukemias progress much more slowly. There are people who are not treated for years once diagnosed with CLL, for example... they don't get treatment until it gets worse, and that can take a very long time for some people.