I wouldn't worry about it because although allopurinol CAN cause hair loss, the odds are very slight that it would. There are very few drugs that absolutely CANNOT cause hair loss as this site attests:
americanhairloss.org/drug_induced_hair_loss
Have you dropped the diuretic? An ACE inhibitor will less likely cause gout and hyperuricemia…IF it works for you and doesn;t cause incessant coughing.
You should be on an analgesic for pain relief…I recommend colchicine along with the allopurinol at least until your pain stops. Give your doctor a call and ask him to call in an Rx.