Let's assume no biological weapons used, just conventional and nukes. From nukes, you have increased cancers of all sorts from fallout and lingering contamination. These cancers will be occurring after a number of years - leukemia and thyroid being most prominent. Anyone with the misfortune to have breathed in any plutonium (not all of it gets used up in the blast) will get lung cancer.
In a full on nuclear war for all intents and purposes all modern medical facilities are gone. Once stocks of medicine have been used up everything from aspirin to antibiotics will be gone. Of course anyone dependent upon medications will be screwed - diabetics go first, people with serious cardiac conditions etc go quickly. Malnutrition will be widespread, both general calorie depletion and deficiency diseases. Kwashiokor, rickets, scurvy, beri-beri etc will reappear and malnourished folks have diminished immune response. Children will show the worst effects. Infectious diseases, especially those associated with poor sanitation and unsafe water such as cholera, typhoid, various diarrheal diseases. Poor sanitation no washing you'll see typhus, Rocky Mountain Spotted Fever & other tick borne diseases. Bubonic plague has reservoirs (four corners area of American southwest in prairie dogs and other rodents for example) and this will jump back to flea infested humans. Common place infections such as boils or cellulitis will become potentially fatal without antibiotics.
To the extent you have doctors/nurses surviving and some medical texts and enough organization you can ameliorate this with enforcing sanitation, using effective natural remedies or food supplements. Eventually some basic production of medicines could be restarted. Chloroform and ether are easy to produce, and with those and basic clean/aseptic/antiseptic technique some surgery could be done safely - of course infections, shock (no IVs or blood transfusions).
Basically you go back to early 19th century medical care, although knowledge about germ theory, sanitation, surgery, anatomy is out there in pockets. Infectious disease will sweep through a population with weakened immune systems from radiation or malnutrition or both.