If you think it's gapeworm, worm them with Fenbendazole or Valbazen or Flubenvet.
From your discription, sounds more likely a respiratory infection. Could be viral, fungal or bacterial. Antibiotics will work only if it's bacterial. Viral infections are not treatable with antibiotics.
Possible causes: Asperillosis, Mycoplasma gallisepticum, Coryza or Infectious bronchitis. These can be treated with Tylan 50, Baytril, and Sulfadimethoxine.
Pheasants are more susceptible to gapeworm than other fowl. Usually picked up from ingesting earthworms, feces of infected birds, contaminated waterers and snails.
Respiratory and fungal infections are usually associated with wet conditions in their enclosure.