But I should be interested to know the dimensions of their divergence (severity?, subjective vs. objective?, popular vs. scientific?).

Interesting question. Short of looking them up in a dictionary or googling for different uses of the words online, here's my subjective and personal observations. For me, disease seems a more neutral and medical term, while illness and sickness tie for less medical and more popular. This probably has more to do with the etymologies of the words. Sick and ill are current adjectives descrbing the patient's state, but though ease is a word, it's a noun and doesn't describe a state. Also, illness seems the least medical of the trio, because of it's older connection with evil (ill as in bad). You used a fourth term in your subject, condition, but there's also lesser words for diseases like ailment, affliction, indisposition, and malady. You can write a book about diseases and illnesses, but not as easily about sicknesses, but you may have a sickness unto death. There is sleeping sickness, but that may be more for the alliteration.

What are the differences between Krankheit, Erkrankung, and Leiden?