What was wrong with Presea Testarossa?
Yes, whatever happened with the reactor probably caused it.  I don't think radiation poisoning is the exact condition, if only because an advanced, experimental technomagical mana reactor shouldn't be using nuclear fission, but the symptoms and progression seem to match media portrayals of that — and of tuberculosis, more closely; the "beautiful pallor", the exhaustion, and coughing up blood were originally markers, in prose and poetry, then later in film, of what was generally referred to as "consumption".  Tuberculosis is no longer a major killer in the developed world, but its symptoms remain in media as the go-to nonspecific slow, wasting death.

As for the time factor, I think her disease was the determiner there; Alicia wasn't going to get any more dead.  Whatever the actual cause of her illness, I believe that it was exacerbated by her throwing herself into Project Fate; as you say, she spent 22 years on that before Fate herself was decanted, presumably working at all hours and neglecting her health; she probably knew she was dying slowly before she realized Fate wasn't Alicia and moved to her mad attempt to reach Al-Hazard.  She gave Linith a hard (and near-impossible) deadline by which to build Bardiche and train Fate as a top-flight combat mage, so she certainly knew she had little time by then.